[
  {
    "task_name": "QKP",
    "problem_type": "QKP",
    "instruction": "Many people know that packing for a hike isn’t just about individual items—each piece has its own weight and usefulness, and some two-item combos give an extra perk when both are carried. The task here is to select a subset of those unique items so the total usefulness (sum of the usefulness values of the selected items plus any extra bonuses for specific pairs that are both in the pack) is as big as possible, while keeping the total weight inside the backpack’s limit. Items are singletons—can’t take duplicates or fractions. The exact items, weights, usefulness scores, and pair bonuses are shown below.\n\nBelow are the 21 available items (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21) and the backpack weight limit 29308.\n\n| gear_id | base_usefulness |\n|---|---|\n| 1 | 3088 |\n| 2 | 5301 |\n| 3 | 4441 |\n| 4 | 393 |\n| 5 | 5478 |\n| 6 | 6001 |\n| 7 | 5974 |\n| 8 | 4107 |\n| 9 | 3437 |\n| 10 | 2609 |\n| 11 | 2557 |\n| 12 | 487 |\n| 13 | 729 |\n| 14 | 480 |\n| 15 | 6049 |\n| 16 | 3798 |\n| 17 | 4514 |\n| 18 | 6432 |\n| 19 | 1475 |\n| 20 | 6481 |\n| 21 | 963 |\n\n| gear_id | item_weight |\n|---|---|\n| 1 | 713 |\n| 2 | 2461 |\n| 3 | 5656 |\n| 4 | 3265 |\n| 5 | 5832 |\n| 6 | 3638 |\n| 7 | 960 |\n| 8 | 6311 |\n| 9 | 1663 |\n| 10 | 3973 |\n| 11 | 2118 |\n| 12 | 672 |\n| 13 | 2144 |\n| 14 | 1411 |\n| 15 | 2489 |\n| 16 | 3373 |\n| 17 | 1756 |\n| 18 | 225 |\n| 19 | 4157 |\n| 20 | 6260 |\n| 21 | 4962 |\n\n| gear_i_id | gear_j_id | pair_bonus |\n|---|---|---|\n| 1 | 2 | 1364 |\n| 1 | 3 | 1131 |\n| 1 | 4 | 2027 |\n| 1 | 5 | 1610 |\n| 1 | 6 | 1274 |\n| 1 | 7 | 1451 |\n| 1 | 8 | 1382 |\n| 1 | 9 | 1855 |\n| 1 | 10 | 669 |\n| 1 | 11 | 1110 |\n| 1 | 12 | 1687 |\n| 1 | 13 | 1253 |\n| 1 | 14 | 462 |\n| 1 | 15 | 1667 |\n| 1 | 16 | 1073 |\n| 1 | 17 | 400 |\n| 1 | 18 | 1677 |\n| 1 | 19 | 1958 |\n| 1 | 20 | 1801 |\n| 1 | 21 | 4037 |\n| 2 | 3 | 3007 |\n| 2 | 4 | 3661 |\n| 2 | 5 | 1048 |\n| 2 | 6 | 1618 |\n| 2 | 7 | 1768 |\n| 2 | 8 | 2453 |\n| 2 | 9 | 3948 |\n| 2 | 10 | 3969 |\n| 2 | 11 | 2660 |\n| 2 | 12 | 3619 |\n| 2 | 13 | 2227 |\n| 2 | 14 | 2844 |\n| 2 | 15 | 3683 |\n| 2 | 16 | 744 |\n| 2 | 17 | 2137 |\n| 2 | 18 | 3901 |\n| 2 | 19 | 3153 |\n| 2 | 20 | 4611 |\n| 2 | 21 | 4812 |\n| 3 | 4 | 2078 |\n| 3 | 5 | 1674 |\n| 3 | 6 | 2914 |\n| 3 | 7 | 5323 |\n| 3 | 8 | 5225 |\n| 3 | 9 | 5470 |\n| 3 | 10 | 2481 |\n| 3 | 11 | 4747 |\n| 3 | 12 | 3482 |\n| 3 | 13 | 2278 |\n| 3 | 14 | 2644 |\n| 3 | 15 | 5155 |\n| 3 | 16 | 1779 |\n| 3 | 17 | 5751 |\n| 3 | 18 | 1456 |\n| 3 | 19 | 4925 |\n| 3 | 20 | 2028 |\n| 3 | 21 | 1911 |\n| 4 | 5 | 4111 |\n| 4 | 6 | 2320 |\n| 4 | 7 | 2751 |\n| 4 | 8 | 4212 |\n| 4 | 9 | 1530 |\n| 4 | 10 | 3030 |\n| 4 | 11 | 4018 |\n| 4 | 12 | 2888 |\n| 4 | 13 | 3832 |\n| 4 | 14 | 3976 |\n| 4 | 15 | 4493 |\n| 4 | 16 | 3684 |\n| 4 | 17 | 3202 |\n| 4 | 18 | 3631 |\n| 4 | 19 | 3854 |\n| 4 | 20 | 3322 |\n| 4 | 21 | 3789 |\n| 5 | 6 | 3234 |\n| 5 | 7 | 5267 |\n| 5 | 8 | 5554 |\n| 5 | 9 | 5348 |\n| 5 | 10 | 3745 |\n| 5 | 11 | 2196 |\n| 5 | 12 | 4320 |\n| 5 | 13 | 2862 |\n| 5 | 14 | 4969 |\n| 5 | 15 | 3769 |\n| 5 | 16 | 5458 |\n| 5 | 17 | 4632 |\n| 5 | 18 | 4121 |\n| 5 | 19 | 3427 |\n| 5 | 20 | 2946 |\n| 5 | 21 | 4476 |\n| 6 | 7 | 4801 |\n| 6 | 8 | 2714 |\n| 6 | 9 | 4203 |\n| 6 | 10 | 4242 |\n| 6 | 11 | 4400 |\n| 6 | 12 | 1827 |\n| 6 | 13 | 3018 |\n| 6 | 14 | 4772 |\n| 6 | 15 | 3173 |\n| 6 | 16 | 3322 |\n| 6 | 17 | 3255 |\n| 6 | 18 | 2864 |\n| 6 | 19 | 4958 |\n| 6 | 20 | 4941 |\n| 6 | 21 | 5100 |\n| 7 | 8 | 1939 |\n| 7 | 9 | 1425 |\n| 7 | 10 | 353 |\n| 7 | 11 | 577 |\n| 7 | 12 | 2271 |\n| 7 | 13 | 2337 |\n| 7 | 14 | 1928 |\n| 7 | 15 | 1879 |\n| 7 | 16 | 1932 |\n| 7 | 17 | 2089 |\n| 7 | 18 | 1625 |\n| 7 | 19 | 1956 |\n| 7 | 20 | 1307 |\n| 7 | 21 | 571 |\n| 8 | 9 | 4937 |\n| 8 | 10 | 3947 |\n| 8 | 11 | 6137 |\n| 8 | 12 | 5613 |\n| 8 | 13 | 2614 |\n| 8 | 14 | 5121 |\n| 8 | 15 | 2929 |\n| 8 | 16 | 2393 |\n| 8 | 17 | 3747 |\n| 8 | 18 | 1837 |\n| 8 | 19 | 5176 |\n| 8 | 20 | 4430 |\n| 8 | 21 | 1978 |\n| 9 | 10 | 2696 |\n| 9 | 11 | 2990 |\n| 9 | 12 | 1639 |\n| 9 | 13 | 3121 |\n| 9 | 14 | 2902 |\n| 9 | 15 | 3913 |\n| 9 | 16 | 977 |\n| 9 | 17 | 1172 |\n| 9 | 18 | 2015 |\n| 9 | 19 | 1422 |\n| 9 | 20 | 3009 |\n| 9 | 21 | 1470 |\n| 10 | 11 | 1491 |\n| 10 | 12 | 4626 |\n| 10 | 13 | 956 |\n| 10 | 14 | 2406 |\n| 10 | 15 | 2309 |\n| 10 | 16 | 1605 |\n| 10 | 17 | 2987 |\n| 10 | 18 | 2156 |\n| 10 | 19 | 1641 |\n| 10 | 20 | 1719 |\n| 10 | 21 | 2572 |\n| 11 | 12 | 3357 |\n| 11 | 13 | 2158 |\n| 11 | 14 | 1881 |\n| 11 | 15 | 1296 |\n| 11 | 16 | 3362 |\n| 11 | 17 | 4387 |\n| 11 | 18 | 1769 |\n| 11 | 19 | 4934 |\n| 11 | 20 | 5262 |\n| 11 | 21 | 4445 |\n| 12 | 13 | 3852 |\n| 12 | 14 | 4168 |\n| 12 | 15 | 871 |\n| 12 | 16 | 4372 |\n| 12 | 17 | 3950 |\n| 12 | 18 | 3747 |\n| 12 | 19 | 4242 |\n| 12 | 20 | 1382 |\n| 12 | 21 | 216 |\n| 13 | 14 | 4542 |\n| 13 | 15 | 2877 |\n| 13 | 16 | 2073 |\n| 13 | 17 | 704 |\n| 13 | 18 | 1373 |\n| 13 | 19 | 1315 |\n| 13 | 20 | 1693 |\n| 13 | 21 | 2931 |\n| 14 | 15 | 4925 |\n| 14 | 16 | 1700 |\n| 14 | 17 | 5495 |\n| 14 | 18 | 1391 |\n| 14 | 19 | 4706 |\n| 14 | 20 | 608 |\n| 14 | 21 | 1010 |\n| 15 | 16 | 3083 |\n| 15 | 17 | 4446 |\n| 15 | 18 | 4091 |\n| 15 | 19 | 1616 |\n| 15 | 20 | 1984 |\n| 15 | 21 | 2545 |\n| 16 | 17 | 2169 |\n| 16 | 18 | 1908 |\n| 16 | 19 | 2946 |\n| 16 | 20 | 2974 |\n| 16 | 21 | 5279 |\n| 17 | 18 | 2267 |\n| 17 | 19 | 2959 |\n| 17 | 20 | 1605 |\n| 17 | 21 | 2702 |\n| 18 | 19 | 1404 |\n| 18 | 20 | 3337 |\n| 18 | 21 | 3199 |\n| 19 | 20 | 2282 |\n| 19 | 21 | 2891 |\n\nChoose a subset that stays within the 29308 limit to maximize total usefulness.\n\nWhen you give the final pick, just drop it into a tiny JSON snippet so it's easy to read and parse, like this:\n\n{\n  \"solution\": [<selected_item_id>, <selected_item_id>, ...]\n}\n\n\"solution\" is simply the list of item IDs you choose to bring in your backpack. Think of each entry in the array as the exact label for one item — that's it. This JSON is just the expected shape (a sketch of the answer format), not the actual solution itself.\n\nAlso, please make sure to use the item identifiers exactly as they appear in the instance input — don't rename them or invent new labels. \n- for example: \"Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.\"",
    "instance": {
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        3088,
        5301,
        4441,
        393,
        5478,
        6001,
        5974,
        4107,
        3437,
        2609,
        2557,
        487,
        729,
        480,
        6049,
        3798,
        4514,
        6432,
        1475,
        6481,
        963
      ],
      "quadratic_coeffs": [
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          1364,
          1131,
          2027,
          1610,
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          1451,
          1382,
          1855,
          669,
          1110,
          1687,
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          1801,
          4037
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        [
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          3661,
          1048,
          1618,
          1768,
          2453,
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          2844,
          3683,
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          2137,
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          3153,
          4611,
          4812
        ],
        [
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          2914,
          5323,
          5225,
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          5155,
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          5751,
          1456,
          4925,
          2028,
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        [
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          4111,
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        ]
      ],
      "weights": [
        713,
        2461,
        5656,
        3265,
        5832,
        3638,
        960,
        6311,
        1663,
        3973,
        2118,
        672,
        2144,
        1411,
        2489,
        3373,
        1756,
        225,
        4157,
        6260,
        4962
      ],
      "capacity": 29308,
      "solution": [
        0,
        1,
        2,
        3,
        5,
        6,
        8,
        10,
        11,
        12,
        13,
        14,
        16,
        17
      ],
      "obj": 303996.0,
      "problem_type": "QKP"
    },
    "solution": [
      0,
      1,
      2,
      3,
      5,
      6,
      8,
      10,
      11,
      12,
      13,
      14,
      16,
      17
    ],
    "obj": 303996.0,
    "instance_variant": {
      "problem_type": "QKP",
      "num_items": 21,
      "capacity": 29308,
      "items": [
        1,
        2,
        3,
        4,
        5,
        6,
        7,
        8,
        9,
        10,
        11,
        12,
        13,
        14,
        15,
        16,
        17,
        18,
        19,
        20,
        21
      ],
      "linear_pairs": [
        {
          "item_id": 1,
          "linear_profit": 3088
        },
        {
          "item_id": 2,
          "linear_profit": 5301
        },
        {
          "item_id": 3,
          "linear_profit": 4441
        },
        {
          "item_id": 4,
          "linear_profit": 393
        },
        {
          "item_id": 5,
          "linear_profit": 5478
        },
        {
          "item_id": 6,
          "linear_profit": 6001
        },
        {
          "item_id": 7,
          "linear_profit": 5974
        },
        {
          "item_id": 8,
          "linear_profit": 4107
        },
        {
          "item_id": 9,
          "linear_profit": 3437
        },
        {
          "item_id": 10,
          "linear_profit": 2609
        },
        {
          "item_id": 11,
          "linear_profit": 2557
        },
        {
          "item_id": 12,
          "linear_profit": 487
        },
        {
          "item_id": 13,
          "linear_profit": 729
        },
        {
          "item_id": 14,
          "linear_profit": 480
        },
        {
          "item_id": 15,
          "linear_profit": 6049
        },
        {
          "item_id": 16,
          "linear_profit": 3798
        },
        {
          "item_id": 17,
          "linear_profit": 4514
        },
        {
          "item_id": 18,
          "linear_profit": 6432
        },
        {
          "item_id": 19,
          "linear_profit": 1475
        },
        {
          "item_id": 20,
          "linear_profit": 6481
        },
        {
          "item_id": 21,
          "linear_profit": 963
        }
      ],
      "weight_pairs": [
        {
          "item_id": 1,
          "weight": 713
        },
        {
          "item_id": 2,
          "weight": 2461
        },
        {
          "item_id": 3,
          "weight": 5656
        },
        {
          "item_id": 4,
          "weight": 3265
        },
        {
          "item_id": 5,
          "weight": 5832
        },
        {
          "item_id": 6,
          "weight": 3638
        },
        {
          "item_id": 7,
          "weight": 960
        },
        {
          "item_id": 8,
          "weight": 6311
        },
        {
          "item_id": 9,
          "weight": 1663
        },
        {
          "item_id": 10,
          "weight": 3973
        },
        {
          "item_id": 11,
          "weight": 2118
        },
        {
          "item_id": 12,
          "weight": 672
        },
        {
          "item_id": 13,
          "weight": 2144
        },
        {
          "item_id": 14,
          "weight": 1411
        },
        {
          "item_id": 15,
          "weight": 2489
        },
        {
          "item_id": 16,
          "weight": 3373
        },
        {
          "item_id": 17,
          "weight": 1756
        },
        {
          "item_id": 18,
          "weight": 225
        },
        {
          "item_id": 19,
          "weight": 4157
        },
        {
          "item_id": 20,
          "weight": 6260
        },
        {
          "item_id": 21,
          "weight": 4962
        }
      ],
      "quadratic_pairs": [
        {
          "item_i_id": 1,
          "item_j_id": 1,
          "quadratic_profit": 1611
        },
        {
          "item_i_id": 1,
          "item_j_id": 2,
          "quadratic_profit": 1364
        },
        {
          "item_i_id": 1,
          "item_j_id": 3,
          "quadratic_profit": 1131
        },
        {
          "item_i_id": 1,
          "item_j_id": 4,
          "quadratic_profit": 2027
        },
        {
          "item_i_id": 1,
          "item_j_id": 5,
          "quadratic_profit": 1610
        },
        {
          "item_i_id": 1,
          "item_j_id": 6,
          "quadratic_profit": 1274
        },
        {
          "item_i_id": 1,
          "item_j_id": 7,
          "quadratic_profit": 1451
        },
        {
          "item_i_id": 1,
          "item_j_id": 8,
          "quadratic_profit": 1382
        },
        {
          "item_i_id": 1,
          "item_j_id": 9,
          "quadratic_profit": 1855
        },
        {
          "item_i_id": 1,
          "item_j_id": 10,
          "quadratic_profit": 669
        },
        {
          "item_i_id": 1,
          "item_j_id": 11,
          "quadratic_profit": 1110
        },
        {
          "item_i_id": 1,
          "item_j_id": 12,
          "quadratic_profit": 1687
        },
        {
          "item_i_id": 1,
          "item_j_id": 13,
          "quadratic_profit": 1253
        },
        {
          "item_i_id": 1,
          "item_j_id": 14,
          "quadratic_profit": 462
        },
        {
          "item_i_id": 1,
          "item_j_id": 15,
          "quadratic_profit": 1667
        },
        {
          "item_i_id": 1,
          "item_j_id": 16,
          "quadratic_profit": 1073
        },
        {
          "item_i_id": 1,
          "item_j_id": 17,
          "quadratic_profit": 400
        },
        {
          "item_i_id": 1,
          "item_j_id": 18,
          "quadratic_profit": 1677
        },
        {
          "item_i_id": 1,
          "item_j_id": 19,
          "quadratic_profit": 1958
        },
        {
          "item_i_id": 1,
          "item_j_id": 20,
          "quadratic_profit": 1801
        },
        {
          "item_i_id": 1,
          "item_j_id": 21,
          "quadratic_profit": 4037
        },
        {
          "item_i_id": 2,
          "item_j_id": 2,
          "quadratic_profit": 3839
        },
        {
          "item_i_id": 2,
          "item_j_id": 3,
          "quadratic_profit": 3007
        },
        {
          "item_i_id": 2,
          "item_j_id": 4,
          "quadratic_profit": 3661
        },
        {
          "item_i_id": 2,
          "item_j_id": 5,
          "quadratic_profit": 1048
        },
        {
          "item_i_id": 2,
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    "instruction": "Someone in the events crew is balancing a menu for a corporate lunch: every dish counts for its own popularity, and certain combinations add bonus appeal when they’re both present. The task is to choose which dishes to include (each dish only once, either included or not) so the combined oven load doesn’t exceed what’s available, and the final guest satisfaction — the sum of individual dish appeals plus any pair bonuses for dishes that appear together — is as large as possible. Concrete details about the dishes, their oven requirements, and the extra pair points follow below.\n\nThey are weighing 22 candidate dishes 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21 against an oven capacity of 5501.\n\n| dish_id | dish_appeal |\n|---|---|\n| 0 | 380 |\n| 1 | 287 |\n| 2 | 330 |\n| 3 | 498 |\n| 4 | 294 |\n| 5 | 359 |\n| 6 | 207 |\n| 7 | 144 |\n| 8 | 130 |\n| 9 | 355 |\n| 10 | 240 |\n| 11 | 310 |\n| 12 | 107 |\n| 13 | 404 |\n| 14 | 152 |\n| 15 | 410 |\n| 16 | 379 |\n| 17 | 399 |\n| 18 | 238 |\n| 19 | 234 |\n| 20 | 248 |\n| 21 | 112 |\n\n| dish_id | oven_load |\n|---|---|\n| 0 | 290 |\n| 1 | 293 |\n| 2 | 354 |\n| 3 | 138 |\n| 4 | 471 |\n| 5 | 163 |\n| 6 | 180 |\n| 7 | 253 |\n| 8 | 189 |\n| 9 | 407 |\n| 10 | 287 |\n| 11 | 222 |\n| 12 | 268 |\n| 13 | 475 |\n| 14 | 366 |\n| 15 | 145 |\n| 16 | 358 |\n| 17 | 419 |\n| 18 | 407 |\n| 19 | 152 |\n| 20 | 142 |\n| 21 | 488 |\n\n| dish_i_id | dish_j_id | pair_bonus_appeal |\n|---|---|---|\n| 0 | 3 | 238 |\n| 0 | 7 | 226 |\n| 0 | 10 | 358 |\n| 0 | 11 | 179 |\n| 0 | 14 | 348 |\n| 0 | 16 | 261 |\n| 1 | 2 | 213 |\n| 1 | 5 | 326 |\n| 1 | 15 | 293 |\n| 1 | 18 | 252 |\n| 2 | 3 | 285 |\n| 2 | 4 | 237 |\n| 2 | 6 | 336 |\n| 2 | 12 | 395 |\n| 2 | 18 | 348 |\n| 3 | 21 | 210 |\n| 4 | 8 | 301 |\n| 4 | 9 | 361 |\n| 4 | 13 | 321 |\n| 5 | 7 | 177 |\n| 5 | 8 | 182 |\n| 5 | 12 | 240 |\n| 5 | 17 | 280 |\n| 5 | 18 | 171 |\n| 5 | 21 | 269 |\n| 6 | 19 | 218 |\n| 6 | 20 | 294 |\n| 7 | 9 | 206 |\n| 7 | 15 | 225 |\n| 8 | 10 | 171 |\n| 8 | 16 | 304 |\n| 9 | 10 | 403 |\n| 9 | 14 | 231 |\n| 9 | 16 | 219 |\n| 9 | 18 | 227 |\n| 9 | 19 | 218 |\n| 10 | 12 | 366 |\n| 10 | 13 | 215 |\n| 10 | 17 | 176 |\n| 10 | 21 | 266 |\n| 11 | 12 | 179 |\n| 12 | 13 | 412 |\n| 12 | 19 | 188 |\n| 12 | 20 | 214 |\n| 13 | 17 | 112 |\n| 14 | 17 | 465 |\n| 15 | 17 | 396 |\n| 15 | 19 | 453 |\n| 16 | 19 | 415 |\n| 18 | 20 | 263 |\n| 20 | 21 | 384 |\n\nThe aim is to select a subset within the 5501 oven limit that maximizes total guest satisfaction across the 22 dishes.\n\nOh, and when you send back your choice, please stick to this simple JSON layout so it's easy to read and process:\n\n{\n  \"solution\": [<selected_dish_id>, <selected_dish_id>, ...]\n}\n\n\"solution\" is just a list of the dishes you decide to include — each entry is the exact dish identifier from the instance (one per chosen dish). Think of the placeholders like little checkboxes: replace each <selected_dish_id> with the ID of a dish you want on the menu. This block is just a sketch of the shape we expect, not the actual answer.\n\nPlease use the identifiers exactly as they appear in the instance input — do not rename them or invent new labels. For example: \"Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.\"",
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          "quadratic_profit": 215
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          "item_i_id": 10,
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          "item_j_id": 12,
          "quadratic_profit": 179
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          "item_i_id": 12,
          "item_j_id": 13,
          "quadratic_profit": 412
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          "item_i_id": 12,
          "item_j_id": 19,
          "quadratic_profit": 188
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          "item_i_id": 12,
          "item_j_id": 20,
          "quadratic_profit": 214
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          "quadratic_profit": 213
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          "quadratic_profit": 112
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          "item_j_id": 14,
          "quadratic_profit": 213
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          "quadratic_profit": 465
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          "quadratic_profit": 396
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          "item_j_id": 19,
          "quadratic_profit": 453
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          "quadratic_profit": 415
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          "item_j_id": 20,
          "quadratic_profit": 263
        },
        {
          "item_i_id": 20,
          "item_j_id": 21,
          "quadratic_profit": 384
        }
      ]
    },
    "solution_variant": [
      0,
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      4,
      5,
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      7,
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      10,
      12,
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    "context_index": 2,
    "input_format": "markdown_table",
    "input_index_base": 0
  },
  {
    "task_name": "QKP",
    "problem_type": "QKP",
    "instruction": "Recently there was a move where the problem was choosing which boxes to cram into a single car so the trip hauled the most value. Each box brings its own value and uses up some of the trunk’s capacity, and some specific pairs of boxes add extra value when they’re together, so what’s counted is the sum of individual box values plus any bonuses from those pairs. Since the trunk can’t be overloaded and every box is unique (no duplicates), the aim is to pick the combination that yields the highest total worth without exceeding the weight allowance. The detailed list of boxes, weights, values, and pairwise bonuses follows below.\n\n{\n  \"total_boxes\": 25,\n  \"trunk_weight_capacity\": 1068,\n  \"box_ids\": [\n    \"A\",\n    \"B\",\n    \"C\",\n    \"D\",\n    \"E\",\n    \"F\",\n    \"G\",\n    \"H\",\n    \"I\",\n    \"J\",\n    \"K\",\n    \"L\",\n    \"M\",\n    \"N\",\n    \"O\",\n    \"P\",\n    \"Q\",\n    \"R\",\n    \"S\",\n    \"T\",\n    \"U\",\n    \"V\",\n    \"W\",\n    \"X\",\n    \"Y\"\n  ],\n  \"linear\": [\n    {\n      \"box_id\": \"A\",\n      \"box_value\": 105\n    },\n    {\n      \"box_id\": \"B\",\n      \"box_value\": 98\n    },\n    {\n      \"box_id\": \"C\",\n      \"box_value\": 100\n    },\n    {\n      \"box_id\": \"D\",\n      \"box_value\": 101\n    },\n    {\n      \"box_id\": \"E\",\n      \"box_value\": 102\n    },\n    {\n      \"box_id\": \"F\",\n      \"box_value\": 97\n    },\n    {\n      \"box_id\": \"G\",\n      \"box_value\": 104\n    },\n    {\n      \"box_id\": \"H\",\n      \"box_value\": 100\n    },\n    {\n      \"box_id\": \"I\",\n      \"box_value\": 99\n    },\n    {\n      \"box_id\": \"J\",\n      \"box_value\": 99\n    },\n    {\n      \"box_id\": \"K\",\n      \"box_value\": 110\n    },\n    {\n      \"box_id\": \"L\",\n      \"box_value\": 101\n    },\n    {\n      \"box_id\": \"M\",\n      \"box_value\": 98\n    },\n    {\n      \"box_id\": \"N\",\n      \"box_value\": 95\n    },\n    {\n      \"box_id\": \"O\",\n      \"box_value\": 99\n    },\n    {\n      \"box_id\": \"P\",\n      \"box_value\": 108\n    },\n    {\n      \"box_id\": \"Q\",\n      \"box_value\": 98\n    },\n    {\n      \"box_id\": \"R\",\n      \"box_value\": 107\n    },\n    {\n      \"box_id\": \"S\",\n      \"box_value\": 104\n    },\n    {\n      \"box_id\": \"T\",\n      \"box_value\": 108\n    },\n    {\n      \"box_id\": \"U\",\n      \"box_value\": 100\n    },\n    {\n      \"box_id\": \"V\",\n      \"box_value\": 98\n    },\n    {\n      \"box_id\": \"W\",\n      \"box_value\": 105\n    },\n    {\n      \"box_id\": \"X\",\n      \"box_value\": 98\n    },\n    {\n      \"box_id\": \"Y\",\n      \"box_value\": 100\n    }\n  ],\n  \"weights\": [\n    {\n      \"box_id\": \"A\",\n      \"box_weight\": 105\n    },\n    {\n      \"box_id\": \"B\",\n      \"box_weight\": 103\n    },\n    {\n      \"box_id\": \"C\",\n      \"box_weight\": 102\n    },\n    {\n      \"box_id\": \"D\",\n      \"box_weight\": 101\n    },\n    {\n      \"box_id\": \"E\",\n      \"box_weight\": 103\n    },\n    {\n      \"box_id\": \"F\",\n      \"box_weight\": 101\n    },\n    {\n      \"box_id\": \"G\",\n      \"box_weight\": 103\n    },\n    {\n      \"box_id\": \"H\",\n      \"box_weight\": 102\n    },\n    {\n      \"box_id\": \"I\",\n      \"box_weight\": 102\n    },\n    {\n      \"box_id\": \"J\",\n      \"box_weight\": 101\n    },\n    {\n      \"box_id\": \"K\",\n      \"box_weight\": 105\n    },\n    {\n      \"box_id\": \"L\",\n      \"box_weight\": 101\n    },\n    {\n      \"box_id\": \"M\",\n      \"box_weight\": 102\n    },\n    {\n      \"box_id\": \"N\",\n      \"box_weight\": 100\n    },\n    {\n      \"box_id\": \"O\",\n      \"box_weight\": 100\n    },\n    {\n      \"box_id\": \"P\",\n      \"box_weight\": 104\n    },\n    {\n      \"box_id\": \"Q\",\n      \"box_weight\": 103\n    },\n    {\n      \"box_id\": \"R\",\n      \"box_weight\": 102\n    },\n    {\n      \"box_id\": \"S\",\n      \"box_weight\": 104\n    },\n    {\n      \"box_id\": \"T\",\n      \"box_weight\": 104\n    },\n    {\n      \"box_id\": \"U\",\n      \"box_weight\": 105\n    },\n    {\n      \"box_id\": \"V\",\n      \"box_weight\": 103\n    },\n    {\n      \"box_id\": \"W\",\n      \"box_weight\": 103\n    },\n    {\n      \"box_id\": \"X\",\n      \"box_weight\": 102\n    },\n    {\n      \"box_id\": \"Y\",\n      \"box_weight\": 101\n    }\n  ],\n  \"quadratic\": [\n    {\n      \"box_i_id\": \"A\",\n      \"box_j_id\": \"D\",\n      \"pair_bonus_value\": 101\n    },\n    {\n      \"box_i_id\": \"A\",\n      \"box_j_id\": \"E\",\n      \"pair_bonus_value\": 101\n    },\n    {\n      \"box_i_id\": \"A\",\n      \"box_j_id\": \"G\",\n      \"pair_bonus_value\": 101\n    },\n    {\n      \"box_i_id\": \"A\",\n      \"box_j_id\": \"H\",\n      \"pair_bonus_value\": 105\n    },\n    {\n      \"box_i_id\": \"A\",\n      \"box_j_id\": \"I\",\n      \"pair_bonus_value\": 101\n    },\n    {\n      \"box_i_id\": \"A\",\n      \"box_j_id\": \"J\",\n      \"pair_bonus_value\": 105\n    },\n    {\n      \"box_i_id\": \"A\",\n      \"box_j_id\": \"K\",\n      \"pair_bonus_value\": 101\n    },\n    {\n      \"box_i_id\": \"A\",\n      \"box_j_id\": \"L\",\n      \"pair_bonus_value\": 104\n    },\n    {\n      \"box_i_id\": \"A\",\n      \"box_j_id\": \"M\",\n      \"pair_bonus_value\": 104\n    },\n    {\n      \"box_i_id\": \"A\",\n      \"box_j_id\": \"N\",\n      \"pair_bonus_value\": 104\n    },\n    {\n      \"box_i_id\": \"A\",\n      \"box_j_id\": \"P\",\n      \"pair_bonus_value\": 104\n    },\n    {\n      \"box_i_id\": \"A\",\n      \"box_j_id\": \"Q\",\n      \"pair_bonus_value\": 104\n    },\n    {\n      \"box_i_id\": \"A\",\n      \"box_j_id\": \"S\",\n      \"pair_bonus_value\": 100\n    },\n    {\n      \"box_i_id\": \"A\",\n      \"box_j_id\": \"T\",\n      \"pair_bonus_value\": 104\n    },\n    {\n      \"box_i_id\": \"A\",\n      \"box_j_id\": \"U\",\n      \"pair_bonus_value\": 103\n    },\n    {\n      \"box_i_id\": \"A\",\n      \"box_j_id\": \"V\",\n      \"pair_bonus_value\": 102\n    },\n    {\n      \"box_i_id\": \"A\",\n      \"box_j_id\": \"W\",\n      \"pair_bonus_value\": 104\n    },\n    {\n      \"box_i_id\": \"A\",\n      \"box_j_id\": \"Y\",\n      \"pair_bonus_value\": 100\n    },\n    {\n      \"box_i_id\": \"B\",\n      \"box_j_id\": \"C\",\n      \"pair_bonus_value\": 103\n    },\n    {\n      \"box_i_id\": \"B\",\n      \"box_j_id\": \"E\",\n      \"pair_bonus_value\": 103\n    },\n  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  \"pair_bonus_value\": 102\n    },\n    {\n      \"box_i_id\": \"I\",\n      \"box_j_id\": \"O\",\n      \"pair_bonus_value\": 102\n    },\n    {\n      \"box_i_id\": \"I\",\n      \"box_j_id\": \"P\",\n      \"pair_bonus_value\": 101\n    },\n    {\n      \"box_i_id\": \"I\",\n      \"box_j_id\": \"Q\",\n      \"pair_bonus_value\": 101\n    },\n    {\n      \"box_i_id\": \"I\",\n      \"box_j_id\": \"R\",\n      \"pair_bonus_value\": 101\n    },\n    {\n      \"box_i_id\": \"I\",\n      \"box_j_id\": \"X\",\n      \"pair_bonus_value\": 101\n    },\n    {\n      \"box_i_id\": \"I\",\n      \"box_j_id\": \"Y\",\n      \"pair_bonus_value\": 101\n    },\n    {\n      \"box_i_id\": \"J\",\n      \"box_j_id\": \"K\",\n      \"pair_bonus_value\": 101\n    },\n    {\n      \"box_i_id\": \"J\",\n      \"box_j_id\": \"L\",\n      \"pair_bonus_value\": 101\n    },\n    {\n      \"box_i_id\": \"J\",\n      \"box_j_id\": \"M\",\n      \"pair_bonus_value\": 101\n    },\n    {\n      \"box_i_id\": \"J\",\n      \"box_j_id\": \"N\",\n      \"pair_bonus_value\": 101\n    },\n    {\n      \"box_i_id\": \"J\",\n      \"box_j_id\": \"O\",\n      \"pair_bonus_value\": 104\n    },\n    {\n      \"box_i_id\": \"J\",\n      \"box_j_id\": \"Q\",\n      \"pair_bonus_value\": 101\n    },\n    {\n      \"box_i_id\": \"J\",\n      \"box_j_id\": \"R\",\n      \"pair_bonus_value\": 104\n    },\n    {\n      \"box_i_id\": \"J\",\n      \"box_j_id\": \"S\",\n      \"pair_bonus_value\": 104\n    },\n    {\n      \"box_i_id\": \"J\",\n      \"box_j_id\": \"T\",\n      \"pair_bonus_value\": 102\n    },\n    {\n      \"box_i_id\": \"J\",\n      \"box_j_id\": \"U\",\n      \"pair_bonus_value\": 102\n    },\n    {\n      \"box_i_id\": \"J\",\n      \"box_j_id\": \"V\",\n      \"pair_bonus_value\": 101\n    },\n    {\n      \"box_i_id\": \"J\",\n      \"box_j_id\": \"W\",\n      \"pair_bonus_value\": 104\n    },\n    {\n      \"box_i_id\": \"J\",\n      \"box_j_id\": \"X\",\n      \"pair_bonus_value\": 105\n    },\n    {\n      \"box_i_id\": \"J\",\n      \"box_j_id\": \"Y\",\n      \"pair_bonus_value\": 100\n    },\n    {\n      \"box_i_id\": \"K\",\n      \"box_j_id\": \"L\",\n      \"pair_bonus_value\": 100\n    },\n    {\n      \"box_i_id\": \"K\",\n      \"box_j_id\": \"M\",\n      \"pair_bonus_value\": 103\n    },\n    {\n      \"box_i_id\": \"K\",\n      \"box_j_id\": \"N\",\n      \"pair_bonus_value\": 101\n    },\n    {\n      \"box_i_id\": \"K\",\n      \"box_j_id\": \"O\",\n      \"pair_bonus_value\": 104\n    },\n    {\n      \"box_i_id\": \"K\",\n      \"box_j_id\": \"P\",\n      \"pair_bonus_value\": 100\n    },\n    {\n      \"box_i_id\": \"K\",\n      \"box_j_id\": \"Q\",\n      \"pair_bonus_value\": 103\n    },\n    {\n      \"box_i_id\": \"K\",\n      \"box_j_id\": \"R\",\n      \"pair_bonus_value\": 100\n    },\n    {\n      \"box_i_id\": \"K\",\n      \"box_j_id\": \"S\",\n      \"pair_bonus_value\": 100\n    },\n    {\n      \"box_i_id\": \"K\",\n      \"box_j_id\": \"T\",\n      \"pair_bonus_value\": 100\n    },\n    {\n      \"box_i_id\": \"K\",\n      \"box_j_id\": \"V\",\n      \"pair_bonus_value\": 104\n    },\n    {\n      \"box_i_id\": \"K\",\n      \"box_j_id\": \"W\",\n      \"pair_bonus_value\": 104\n    },\n    {\n      \"box_i_id\": \"K\",\n      \"box_j_id\": \"X\",\n      \"pair_bonus_value\": 100\n    },\n    {\n      \"box_i_id\": \"K\",\n      \"box_j_id\": \"Y\",\n      \"pair_bonus_value\": 100\n    },\n    {\n      \"box_i_id\": \"L\",\n      \"box_j_id\": \"N\",\n      \"pair_bonus_value\": 104\n    },\n    {\n      \"box_i_id\": \"L\",\n      \"box_j_id\": \"O\",\n      \"pair_bonus_value\": 103\n    },\n    {\n      \"box_i_id\": \"L\",\n      \"box_j_id\": \"Q\",\n      \"pair_bonus_value\": 103\n    },\n    {\n      \"box_i_id\": \"L\",\n      \"box_j_id\": \"R\",\n      \"pair_bonus_value\": 104\n    },\n    {\n      \"box_i_id\": \"L\",\n      \"box_j_id\": \"S\",\n      \"pair_bonus_value\": 102\n    },\n  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  \"pair_bonus_value\": 102\n    },\n    {\n      \"box_i_id\": \"M\",\n      \"box_j_id\": \"X\",\n      \"pair_bonus_value\": 101\n    },\n    {\n      \"box_i_id\": \"N\",\n      \"box_j_id\": \"P\",\n      \"pair_bonus_value\": 100\n    },\n    {\n      \"box_i_id\": \"N\",\n      \"box_j_id\": \"S\",\n      \"pair_bonus_value\": 102\n    },\n    {\n      \"box_i_id\": \"N\",\n      \"box_j_id\": \"T\",\n      \"pair_bonus_value\": 103\n    },\n    {\n      \"box_i_id\": \"N\",\n      \"box_j_id\": \"U\",\n      \"pair_bonus_value\": 104\n    },\n    {\n      \"box_i_id\": \"N\",\n      \"box_j_id\": \"V\",\n      \"pair_bonus_value\": 100\n    },\n    {\n      \"box_i_id\": \"N\",\n      \"box_j_id\": \"W\",\n      \"pair_bonus_value\": 104\n    },\n    {\n      \"box_i_id\": \"N\",\n      \"box_j_id\": \"X\",\n      \"pair_bonus_value\": 100\n    },\n    {\n      \"box_i_id\": \"N\",\n      \"box_j_id\": \"Y\",\n      \"pair_bonus_value\": 104\n    },\n    {\n      \"box_i_id\": \"O\",\n      \"box_j_id\": \"P\",\n      \"pair_bonus_value\": 100\n    },\n    {\n      \"box_i_id\": \"O\",\n      \"box_j_id\": \"Q\",\n      \"pair_bonus_value\": 101\n    },\n    {\n      \"box_i_id\": \"O\",\n      \"box_j_id\": \"R\",\n      \"pair_bonus_value\": 100\n    },\n    {\n      \"box_i_id\": \"O\",\n      \"box_j_id\": \"S\",\n      \"pair_bonus_value\": 100\n    },\n    {\n      \"box_i_id\": \"O\",\n      \"box_j_id\": \"T\",\n      \"pair_bonus_value\": 101\n    },\n    {\n      \"box_i_id\": \"O\",\n      \"box_j_id\": \"U\",\n      \"pair_bonus_value\": 103\n    },\n    {\n      \"box_i_id\": \"O\",\n      \"box_j_id\": \"V\",\n      \"pair_bonus_value\": 102\n    },\n    {\n      \"box_i_id\": \"O\",\n      \"box_j_id\": \"W\",\n      \"pair_bonus_value\": 104\n    },\n    {\n      \"box_i_id\": \"O\",\n      \"box_j_id\": \"X\",\n      \"pair_bonus_value\": 104\n    },\n    {\n      \"box_i_id\": \"O\",\n      \"box_j_id\": \"Y\",\n      \"pair_bonus_value\": 104\n    },\n    {\n      \"box_i_id\": \"P\",\n      \"box_j_id\": \"Q\",\n      \"pair_bonus_value\": 104\n    },\n    {\n      \"box_i_id\": \"P\",\n      \"box_j_id\": \"R\",\n      \"pair_bonus_value\": 100\n    },\n    {\n      \"box_i_id\": \"P\",\n      \"box_j_id\": \"T\",\n      \"pair_bonus_value\": 101\n    },\n    {\n      \"box_i_id\": \"P\",\n      \"box_j_id\": \"V\",\n      \"pair_bonus_value\": 103\n    },\n    {\n      \"box_i_id\": \"P\",\n      \"box_j_id\": \"W\",\n      \"pair_bonus_value\": 100\n    },\n    {\n      \"box_i_id\": \"P\",\n      \"box_j_id\": \"X\",\n      \"pair_bonus_value\": 103\n    },\n    {\n      \"box_i_id\": \"P\",\n      \"box_j_id\": \"Y\",\n      \"pair_bonus_value\": 100\n    },\n    {\n      \"box_i_id\": \"Q\",\n      \"box_j_id\": \"R\",\n      \"pair_bonus_value\": 103\n    },\n    {\n      \"box_i_id\": \"Q\",\n      \"box_j_id\": \"S\",\n      \"pair_bonus_value\": 103\n    },\n    {\n      \"box_i_id\": \"Q\",\n      \"box_j_id\": \"T\",\n      \"pair_bonus_value\": 100\n    },\n    {\n      \"box_i_id\": \"Q\",\n      \"box_j_id\": \"U\",\n      \"pair_bonus_value\": 101\n    },\n    {\n      \"box_i_id\": \"Q\",\n      \"box_j_id\": \"V\",\n      \"pair_bonus_value\": 104\n    },\n    {\n      \"box_i_id\": \"Q\",\n      \"box_j_id\": \"X\",\n      \"pair_bonus_value\": 104\n    },\n    {\n      \"box_i_id\": \"Q\",\n      \"box_j_id\": \"Y\",\n      \"pair_bonus_value\": 103\n    },\n    {\n      \"box_i_id\": \"R\",\n      \"box_j_id\": \"T\",\n      \"pair_bonus_value\": 100\n    },\n    {\n      \"box_i_id\": \"R\",\n      \"box_j_id\": \"V\",\n      \"pair_bonus_value\": 100\n    },\n    {\n      \"box_i_id\": \"R\",\n      \"box_j_id\": \"W\",\n      \"pair_bonus_value\": 100\n    },\n    {\n      \"box_i_id\": \"R\",\n      \"box_j_id\": \"X\",\n      \"pair_bonus_value\": 100\n    },\n    {\n      \"box_i_id\": \"R\",\n      \"box_j_id\": \"Y\",\n      \"pair_bonus_value\": 100\n    },\n    {\n      \"box_i_id\": \"S\",\n      \"box_j_id\": \"T\",\n      \"pair_bonus_value\": 104\n    },\n    {\n      \"box_i_id\": \"S\",\n      \"box_j_id\": \"V\",\n      \"pair_bonus_value\": 102\n    },\n    {\n      \"box_i_id\": \"S\",\n      \"box_j_id\": \"X\",\n      \"pair_bonus_value\": 101\n    },\n    {\n      \"box_i_id\": \"S\",\n      \"box_j_id\": \"Y\",\n      \"pair_bonus_value\": 100\n    }\n  ]\n}\n\nOh, and when you send back your pick, please stick to a tiny JSON snippet so it's easy to read and validate. It should follow this shape:\n\n{\n  \"solution\": [<selected_item_id>, <selected_item_id>, ...]\n}\n\nHere the \"solution\" array is just a list of the box ids you decided to load into the trunk — one entry per chosen box. This is only a sketch of the expected shape, not the actual selection.\n\nPlease make sure you use the identifiers exactly as they appear in the instance input — do not rename them or invent new labels.\n\nFor example: \"Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.\"",
    "instance": {
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        98,
        100,
        101,
        102,
        97,
        104,
        100,
        99,
        99,
        110,
        101,
        98,
        95,
        99,
        108,
        98,
        107,
        104,
        108,
        100,
        98,
        105,
        98,
        100
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      "quadratic_coeffs": [
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    "problem_type": "QKP",
    "instruction": "Out in the studio someone’s curating a compact show and balancing fun against minutes: choose which songs to play so the entire set feels as enjoyable as possible. Each track contributes a base chunk of enjoyment, and if certain tracks are both included they add a little extra because they transition nicely. The final score is the sum of every chosen song’s enjoyment plus any of those pairwise extras, and the chosen tracks must not exceed the show’s time limit or be repeated. The exact options and timings are given below.\n\n# num_tracks=19\n# total_show_runtime=27264\n# track_ids=A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S\n\ntrack_id,track_enjoyment\nA,343\nB,4088\nC,4451\nD,6001\nE,2843\nF,1289\nG,2557\nH,6003\nI,5120\nJ,5738\nK,291\nL,3132\nM,1995\nN,1182\nO,4111\nP,1937\nQ,4340\nR,5332\nS,6481\n\ntrack_id,track_runtime\nA,527\nB,824\nC,1795\nD,3638\nE,3219\nF,849\nG,2118\nH,6403\nI,359\nJ,1064\nK,3044\nL,5163\nM,1891\nN,1616\nO,5861\nP,5328\nQ,4290\nR,5323\nS,6260\n\ntrack_i_id,track_j_id,transition_bonus\nA,B,1209\nA,C,1753\nA,D,1349\nA,E,1827\nA,F,1551\nA,G,726\nA,H,1561\nA,I,1847\nA,J,1474\nA,K,998\nA,L,1646\nA,M,1344\nA,N,999\nA,O,1640\nA,P,1617\nA,Q,2534\nA,R,3730\nA,S,2153\nB,C,2111\nB,D,1869\nB,E,2088\nB,F,1193\nB,G,2200\nB,H,1668\nB,I,1739\nB,J,1449\nB,K,1436\nB,L,1585\nB,M,1793\nB,N,1064\nB,O,2080\nB,P,5818\nB,Q,1878\nB,R,1962\nB,S,2174\nC,D,2502\nC,E,2003\nC,F,3390\nC,G,2247\nC,H,2602\nC,I,2105\nC,J,1824\nC,K,3156\nC,L,3196\nC,M,1394\nC,N,702\nC,O,2513\nC,P,4420\nC,Q,4689\nC,R,5908\nC,S,4141\nD,E,1832\nD,F,3320\nD,G,4400\nD,H,2540\nD,I,3988\nD,J,4550\nD,K,2411\nD,L,904\nD,M,4617\nD,N,4528\nD,O,4068\nD,P,2535\nD,Q,2006\nD,R,2887\nD,S,4941\nE,F,2547\nE,G,3805\nE,H,4009\nE,I,1867\nE,J,4160\nE,K,4164\nE,L,3716\nE,M,3520\nE,N,2107\nE,O,1069\nE,P,1616\nE,Q,774\nE,R,2784\nE,S,1097\nF,G,1221\nF,H,1997\nF,I,1532\nF,J,2773\nF,K,1831\nF,L,3183\nF,M,1259\nF,N,2225\nF,O,2664\nF,P,1076\nF,Q,2811\nF,R,1991\nF,S,2198\nG,H,891\nG,I,2185\nG,J,1510\nG,K,1652\nG,L,1666\nG,M,2755\nG,N,4325\nG,O,4691\nG,P,4856\nG,Q,4109\nG,R,1547\nG,S,5262\nH,I,5636\nH,J,2974\nH,K,3765\nH,L,1040\nH,M,2790\nH,N,2702\nH,O,2498\nH,P,2678\nH,Q,4007\nH,R,3556\nH,S,3424\nI,J,4633\nI,K,3623\nI,L,3688\nI,M,2540\nI,N,4838\nI,O,4745\nI,P,2658\nI,Q,1410\nI,R,1249\nI,S,2971\nJ,K,2999\nJ,L,2728\nJ,M,3999\nJ,N,2438\nJ,O,2827\nJ,P,1790\nJ,Q,800\nJ,R,2080\nJ,S,3540\nK,L,1128\nK,M,1502\nK,N,1047\nK,O,1781\nK,P,1912\nK,Q,966\nK,R,1626\nK,S,924\nL,M,2018\nL,N,3431\nL,O,3594\nL,P,4203\nL,Q,1859\nL,R,4050\nL,S,1976\nM,N,2455\nM,O,3903\nM,P,4843\nM,Q,5409\nM,R,5301\nM,S,4555\nN,O,2646\nN,P,3163\nN,Q,2677\nN,R,2998\nN,S,2000\nO,P,4130\nO,Q,4201\nO,R,4448\nO,S,4046\n\nAlso, when you hand in your final playlist, it'd be handy if you put the chosen tracks into a tiny JSON snippet so it's easy to read by whoever's cuing the set. Something like this will do:\n\n{\n  \"solution\": [<selected_track_id>, <selected_track_id>, ...]\n}\n\nPretty simple: \"solution\" is the list of tracks you plan to play (in any order). Think of each array entry as the exact track identifier from the instance — that's the only thing the scheduler needs from you. This JSON is just a sketch of the shape I expect, not your actual answer.\n\nPlease make sure every identifier you use matches the instance exactly — no renaming, no made-up labels. For example:\n- \"Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.\"",
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          "item_i_id": "I",
          "item_j_id": "O",
          "quadratic_profit": 4745
        },
        {
          "item_i_id": "I",
          "item_j_id": "P",
          "quadratic_profit": 2658
        },
        {
          "item_i_id": "I",
          "item_j_id": "Q",
          "quadratic_profit": 1410
        },
        {
          "item_i_id": "I",
          "item_j_id": "R",
          "quadratic_profit": 1249
        },
        {
          "item_i_id": "I",
          "item_j_id": "S",
          "quadratic_profit": 2971
        },
        {
          "item_i_id": "J",
          "item_j_id": "J",
          "quadratic_profit": 978
        },
        {
          "item_i_id": "J",
          "item_j_id": "K",
          "quadratic_profit": 2999
        },
        {
          "item_i_id": "J",
          "item_j_id": "L",
          "quadratic_profit": 2728
        },
        {
          "item_i_id": "J",
          "item_j_id": "M",
          "quadratic_profit": 3999
        },
        {
          "item_i_id": "J",
          "item_j_id": "N",
          "quadratic_profit": 2438
        },
        {
          "item_i_id": "J",
          "item_j_id": "O",
          "quadratic_profit": 2827
        },
        {
          "item_i_id": "J",
          "item_j_id": "P",
          "quadratic_profit": 1790
        },
        {
          "item_i_id": "J",
          "item_j_id": "Q",
          "quadratic_profit": 800
        },
        {
          "item_i_id": "J",
          "item_j_id": "R",
          "quadratic_profit": 2080
        },
        {
          "item_i_id": "J",
          "item_j_id": "S",
          "quadratic_profit": 3540
        },
        {
          "item_i_id": "K",
          "item_j_id": "K",
          "quadratic_profit": 818
        },
        {
          "item_i_id": "K",
          "item_j_id": "L",
          "quadratic_profit": 1128
        },
        {
          "item_i_id": "K",
          "item_j_id": "M",
          "quadratic_profit": 1502
        },
        {
          "item_i_id": "K",
          "item_j_id": "N",
          "quadratic_profit": 1047
        },
        {
          "item_i_id": "K",
          "item_j_id": "O",
          "quadratic_profit": 1781
        },
        {
          "item_i_id": "K",
          "item_j_id": "P",
          "quadratic_profit": 1912
        },
        {
          "item_i_id": "K",
          "item_j_id": "Q",
          "quadratic_profit": 966
        },
        {
          "item_i_id": "K",
          "item_j_id": "R",
          "quadratic_profit": 1626
        },
        {
          "item_i_id": "K",
          "item_j_id": "S",
          "quadratic_profit": 924
        },
        {
          "item_i_id": "L",
          "item_j_id": "L",
          "quadratic_profit": 1361
        },
        {
          "item_i_id": "L",
          "item_j_id": "M",
          "quadratic_profit": 2018
        },
        {
          "item_i_id": "L",
          "item_j_id": "N",
          "quadratic_profit": 3431
        },
        {
          "item_i_id": "L",
          "item_j_id": "O",
          "quadratic_profit": 3594
        },
        {
          "item_i_id": "L",
          "item_j_id": "P",
          "quadratic_profit": 4203
        },
        {
          "item_i_id": "L",
          "item_j_id": "Q",
          "quadratic_profit": 1859
        },
        {
          "item_i_id": "L",
          "item_j_id": "R",
          "quadratic_profit": 4050
        },
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          "item_i_id": "L",
          "item_j_id": "S",
          "quadratic_profit": 1976
        },
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          "item_j_id": "M",
          "quadratic_profit": 2416
        },
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          "item_i_id": "M",
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          "quadratic_profit": 2455
        },
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          "item_j_id": "O",
          "quadratic_profit": 3903
        },
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          "item_i_id": "M",
          "item_j_id": "P",
          "quadratic_profit": 4843
        },
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          "item_i_id": "M",
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          "quadratic_profit": 5409
        },
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          "quadratic_profit": 5301
        },
        {
          "item_i_id": "M",
          "item_j_id": "S",
          "quadratic_profit": 4555
        },
        {
          "item_i_id": "N",
          "item_j_id": "N",
          "quadratic_profit": 2249
        },
        {
          "item_i_id": "N",
          "item_j_id": "O",
          "quadratic_profit": 2646
        },
        {
          "item_i_id": "N",
          "item_j_id": "P",
          "quadratic_profit": 3163
        },
        {
          "item_i_id": "N",
          "item_j_id": "Q",
          "quadratic_profit": 2677
        },
        {
          "item_i_id": "N",
          "item_j_id": "R",
          "quadratic_profit": 2998
        },
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          "item_j_id": "S",
          "quadratic_profit": 2000
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          "quadratic_profit": 4837
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          "item_j_id": "P",
          "quadratic_profit": 4130
        },
        {
          "item_i_id": "O",
          "item_j_id": "Q",
          "quadratic_profit": 4201
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          "item_j_id": "R",
          "quadratic_profit": 4448
        },
        {
          "item_i_id": "O",
          "item_j_id": "S",
          "quadratic_profit": 4046
        }
      ]
    },
    "solution_variant": [
      "A",
      "B",
      "C",
      "D",
      "E",
      "F",
      "G",
      "I",
      "J",
      "K",
      "M",
      "N",
      "S"
    ],
    "context_index": 4,
    "input_format": "csv",
    "input_index_base": "names"
  },
  {
    "task_name": "QKP",
    "problem_type": "QKP",
    "instruction": "Recently a photographer had to trim a massive gear list into one bag for a fast-paced shoot, and the goal was clear: keep things light but versatile. Every lens or accessory adds a basic usefulness score, and some pairings add extra usefulness when both are included, so the final usefulness is calculated by adding each chosen item’s base score and then tacking on any pair bonuses. Weight is the limiter — the bag has a cap — and each item is either taken or not, no duplicates. The exact items, weights, and bonus pairings appear below.\n\nThere are 17 distinct lenses and accessories listed as A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, and the bag's weight capacity is 61323.\nItem A provides a base usefulness score of 526.\nItem B provides a base usefulness score of 1890.\nItem C provides a base usefulness score of 3034.\nItem D provides a base usefulness score of 3869.\nItem E provides a base usefulness score of 5107.\nItem F provides a base usefulness score of 1329.\nItem G provides a base usefulness score of 5553.\nItem H provides a base usefulness score of 474.\nItem I provides a base usefulness score of 5563.\nItem J provides a base usefulness score of 2358.\nItem K provides a base usefulness score of 4160.\nItem L provides a base usefulness score of 5634.\nItem M provides a base usefulness score of 3908.\nItem N provides a base usefulness score of 5834.\nItem O provides a base usefulness score of 2557.\nItem P provides a base usefulness score of 315.\nItem Q provides a base usefulness score of 1737.\nItem A has a weight of 2959.\nItem B has a weight of 5270.\nItem C has a weight of 721.\nItem D has a weight of 5818.\nItem E has a weight of 1018.\nItem F has a weight of 4853.\nItem G has a weight of 4286.\nItem H has a weight of 3635.\nItem I has a weight of 4189.\nItem J has a weight of 755.\nItem K has a weight of 778.\nItem L has a weight of 5660.\nItem M has a weight of 5795.\nItem N has a weight of 3329.\nItem O has a weight of 5435.\nItem P has a weight of 4794.\nItem Q has a weight of 2056.\nIf both A and E are packed, they add an extra usefulness of 2959.\nIf both A and G are packed, they add an extra usefulness of 1050.\nIf both A and L are packed, they add an extra usefulness of 707.\nIf both B and E are packed, they add an extra usefulness of 1691.\nIf both B and F are packed, they add an extra usefulness of 5038.\nIf both B and G are packed, they add an extra usefulness of 4534.\nIf both B and K are packed, they add an extra usefulness of 5270.\nIf both C and D are packed, they add an extra usefulness of 721.\nIf both C and F are packed, they add an extra usefulness of 721.\nIf both C and M are packed, they add an extra usefulness of 721.\nIf both C and N are packed, they add an extra usefulness of 3504.\nIf both D and G are packed, they add an extra usefulness of 3745.\nIf both D and J are packed, they add an extra usefulness of 853.\nIf both D and Q are packed, they add an extra usefulness of 2463.\nIf both E and F are packed, they add an extra usefulness of 1018.\nIf both E and G are packed, they add an extra usefulness of 1018.\nIf both E and J are packed, they add an extra usefulness of 1340.\nIf both E and L are packed, they add an extra usefulness of 1050.\nIf both E and M are packed, they add an extra usefulness of 4853.\nIf both E and O are packed, they add an extra usefulness of 4853.\nIf both E and Q are packed, they add an extra usefulness of 463.\nIf both F and H are packed, they add an extra usefulness of 281.\nIf both F and K are packed, they add an extra usefulness of 2002.\nIf both F and M are packed, they add an extra usefulness of 437.\nIf both F and O are packed, they add an extra usefulness of 3401.\nIf both F and P are packed, they add an extra usefulness of 2875.\nIf both F and Q are packed, they add an extra usefulness of 4435.\nIf both G and O are packed, they add an extra usefulness of 4435.\nIf both H and O are packed, they add an extra usefulness of 540.\nIf both H and Q are packed, they add an extra usefulness of 2169.\nIf both I and J are packed, they add an extra usefulness of 3147.\nIf both I and K are packed, they add an extra usefulness of 2732.\nIf both I and Q are packed, they add an extra usefulness of 334.\nIf both J and K are packed, they add an extra usefulness of 778.\nIf both J and L are packed, they add an extra usefulness of 2761.\nIf both K and M are packed, they add an extra usefulness of 404.\nIf both K and N are packed, they add an extra usefulness of 404.\nIf both K and Q are packed, they add an extra usefulness of 404.\nIf both M and Q are packed, they add an extra usefulness of 367.\nSelections must stay within the 61323 limit to maximize overall usefulness across the 17 candidates.\n\nYou can just reply with the chosen gear in a tiny JSON snippet like this — handy and simple.\n\n{\n  \"solution\": [<gear_id>, <gear_id>, ...]\n}\n\nThink of \"solution\" as the list of items you're packing for the shoot; each <gear_id> is a placeholder for one exact item identifier from the problem instance. This JSON is just a sketch of the shape I want, not the actual selection.\n\nPlease use the identifiers exactly as they appear in the instance input — no renaming and no new labels.\n- for example: \"Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.\"",
    "instance": {
      "linear_coeffs": [
        526,
        1890,
        3034,
        3869,
        5107,
        1329,
        5553,
        474,
        5563,
        2358,
        4160,
        5634,
        3908,
        5834,
        2557,
        315,
        1737
      ],
      "quadratic_coeffs": [
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          0,
          0,
          0,
          2959,
          0,
          1050,
          0,
          0,
          0,
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          0
        ],
        [
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          0,
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          5038,
          4534,
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          5270,
          0,
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          0,
          0,
          0,
          0
        ],
        [
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          721,
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          0,
          0,
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          0,
          0,
          0,
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          3504,
          0,
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        ],
        [
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          0,
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          3745,
          0,
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          853,
          0,
          0,
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        [
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        [
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          281,
          0,
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          2002,
          0,
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        ],
        [
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        [
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        ],
        [
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        [
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        ],
        [
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      ],
      "weights": [
        2959,
        5270,
        721,
        5818,
        1018,
        4853,
        4286,
        3635,
        4189,
        755,
        778,
        5660,
        5795,
        3329,
        5435,
        4794,
        2056
      ],
      "capacity": 61323,
      "solution": [
        0,
        1,
        2,
        3,
        4,
        5,
        6,
        7,
        8,
        9,
        10,
        11,
        12,
        13,
        14,
        16
      ],
      "obj": 131136.0,
      "problem_type": "QKP"
    },
    "solution": [
      0,
      1,
      2,
      3,
      4,
      5,
      6,
      7,
      8,
      9,
      10,
      11,
      12,
      13,
      14,
      16
    ],
    "obj": 131136.0,
    "instance_variant": {
      "problem_type": "QKP",
      "num_items": 17,
      "capacity": 61323,
      "items": [
        "A",
        "B",
        "C",
        "D",
        "E",
        "F",
        "G",
        "H",
        "I",
        "J",
        "K",
        "L",
        "M",
        "N",
        "O",
        "P",
        "Q"
      ],
      "linear_pairs": [
        {
          "item_id": "A",
          "linear_profit": 526
        },
        {
          "item_id": "B",
          "linear_profit": 1890
        },
        {
          "item_id": "C",
          "linear_profit": 3034
        },
        {
          "item_id": "D",
          "linear_profit": 3869
        },
        {
          "item_id": "E",
          "linear_profit": 5107
        },
        {
          "item_id": "F",
          "linear_profit": 1329
        },
        {
          "item_id": "G",
          "linear_profit": 5553
        },
        {
          "item_id": "H",
          "linear_profit": 474
        },
        {
          "item_id": "I",
          "linear_profit": 5563
        },
        {
          "item_id": "J",
          "linear_profit": 2358
        },
        {
          "item_id": "K",
          "linear_profit": 4160
        },
        {
          "item_id": "L",
          "linear_profit": 5634
        },
        {
          "item_id": "M",
          "linear_profit": 3908
        },
        {
          "item_id": "N",
          "linear_profit": 5834
        },
        {
          "item_id": "O",
          "linear_profit": 2557
        },
        {
          "item_id": "P",
          "linear_profit": 315
        },
        {
          "item_id": "Q",
          "linear_profit": 1737
        }
      ],
      "weight_pairs": [
        {
          "item_id": "A",
          "weight": 2959
        },
        {
          "item_id": "B",
          "weight": 5270
        },
        {
          "item_id": "C",
          "weight": 721
        },
        {
          "item_id": "D",
          "weight": 5818
        },
        {
          "item_id": "E",
          "weight": 1018
        },
        {
          "item_id": "F",
          "weight": 4853
        },
        {
          "item_id": "G",
          "weight": 4286
        },
        {
          "item_id": "H",
          "weight": 3635
        },
        {
          "item_id": "I",
          "weight": 4189
        },
        {
          "item_id": "J",
          "weight": 755
        },
        {
          "item_id": "K",
          "weight": 778
        },
        {
          "item_id": "L",
          "weight": 5660
        },
        {
          "item_id": "M",
          "weight": 5795
        },
        {
          "item_id": "N",
          "weight": 3329
        },
        {
          "item_id": "O",
          "weight": 5435
        },
        {
          "item_id": "P",
          "weight": 4794
        },
        {
          "item_id": "Q",
          "weight": 2056
        }
      ],
      "quadratic_pairs": [
        {
          "item_i_id": "A",
          "item_j_id": "E",
          "quadratic_profit": 2959
        },
        {
          "item_i_id": "A",
          "item_j_id": "G",
          "quadratic_profit": 1050
        },
        {
          "item_i_id": "A",
          "item_j_id": "L",
          "quadratic_profit": 707
        },
        {
          "item_i_id": "B",
          "item_j_id": "E",
          "quadratic_profit": 1691
        },
        {
          "item_i_id": "B",
          "item_j_id": "F",
          "quadratic_profit": 5038
        },
        {
          "item_i_id": "B",
          "item_j_id": "G",
          "quadratic_profit": 4534
        },
        {
          "item_i_id": "B",
          "item_j_id": "K",
          "quadratic_profit": 5270
        },
        {
          "item_i_id": "C",
          "item_j_id": "C",
          "quadratic_profit": 721
        },
        {
          "item_i_id": "C",
          "item_j_id": "D",
          "quadratic_profit": 721
        },
        {
          "item_i_id": "C",
          "item_j_id": "F",
          "quadratic_profit": 721
        },
        {
          "item_i_id": "C",
          "item_j_id": "M",
          "quadratic_profit": 721
        },
        {
          "item_i_id": "C",
          "item_j_id": "N",
          "quadratic_profit": 3504
        },
        {
          "item_i_id": "D",
          "item_j_id": "D",
          "quadratic_profit": 2531
        },
        {
          "item_i_id": "D",
          "item_j_id": "G",
          "quadratic_profit": 3745
        },
        {
          "item_i_id": "D",
          "item_j_id": "J",
          "quadratic_profit": 853
        },
        {
          "item_i_id": "D",
          "item_j_id": "Q",
          "quadratic_profit": 2463
        },
        {
          "item_i_id": "E",
          "item_j_id": "F",
          "quadratic_profit": 1018
        },
        {
          "item_i_id": "E",
          "item_j_id": "G",
          "quadratic_profit": 1018
        },
        {
          "item_i_id": "E",
          "item_j_id": "J",
          "quadratic_profit": 1340
        },
        {
          "item_i_id": "E",
          "item_j_id": "L",
          "quadratic_profit": 1050
        },
        {
          "item_i_id": "E",
          "item_j_id": "M",
          "quadratic_profit": 4853
        },
        {
          "item_i_id": "E",
          "item_j_id": "O",
          "quadratic_profit": 4853
        },
        {
          "item_i_id": "E",
          "item_j_id": "Q",
          "quadratic_profit": 463
        },
        {
          "item_i_id": "F",
          "item_j_id": "F",
          "quadratic_profit": 4143
        },
        {
          "item_i_id": "F",
          "item_j_id": "H",
          "quadratic_profit": 281
        },
        {
          "item_i_id": "F",
          "item_j_id": "K",
          "quadratic_profit": 2002
        },
        {
          "item_i_id": "F",
          "item_j_id": "M",
          "quadratic_profit": 437
        },
        {
          "item_i_id": "F",
          "item_j_id": "O",
          "quadratic_profit": 3401
        },
        {
          "item_i_id": "F",
          "item_j_id": "P",
          "quadratic_profit": 2875
        },
        {
          "item_i_id": "F",
          "item_j_id": "Q",
          "quadratic_profit": 4435
        },
        {
          "item_i_id": "G",
          "item_j_id": "G",
          "quadratic_profit": 2590
        },
        {
          "item_i_id": "G",
          "item_j_id": "O",
          "quadratic_profit": 4435
        },
        {
          "item_i_id": "H",
          "item_j_id": "O",
          "quadratic_profit": 540
        },
        {
          "item_i_id": "H",
          "item_j_id": "Q",
          "quadratic_profit": 2169
        },
        {
          "item_i_id": "I",
          "item_j_id": "J",
          "quadratic_profit": 3147
        },
        {
          "item_i_id": "I",
          "item_j_id": "K",
          "quadratic_profit": 2732
        },
        {
          "item_i_id": "I",
          "item_j_id": "Q",
          "quadratic_profit": 334
        },
        {
          "item_i_id": "J",
          "item_j_id": "K",
          "quadratic_profit": 778
        },
        {
          "item_i_id": "J",
          "item_j_id": "L",
          "quadratic_profit": 2761
        },
        {
          "item_i_id": "K",
          "item_j_id": "M",
          "quadratic_profit": 404
        },
        {
          "item_i_id": "K",
          "item_j_id": "N",
          "quadratic_profit": 404
        },
        {
          "item_i_id": "K",
          "item_j_id": "Q",
          "quadratic_profit": 404
        },
        {
          "item_i_id": "L",
          "item_j_id": "L",
          "quadratic_profit": 4511
        },
        {
          "item_i_id": "M",
          "item_j_id": "Q",
          "quadratic_profit": 367
        },
        {
          "item_i_id": "O",
          "item_j_id": "O",
          "quadratic_profit": 281
        }
      ]
    },
    "solution_variant": [
      "A",
      "B",
      "C",
      "D",
      "E",
      "F",
      "G",
      "H",
      "I",
      "J",
      "K",
      "L",
      "M",
      "N",
      "O",
      "Q"
    ],
    "context_index": 5,
    "input_format": "nl",
    "input_index_base": "names"
  },
  {
    "task_name": "QKP",
    "problem_type": "QKP",
    "instruction": "I inherited a small raised bed and now have to pick which plants to put in it. Each plant takes up a bit of root space, and every choice brings a base harvest of veggies or flowers. Some combinations give a little extra — certain pairs are better buddies and boost each other’s output when they’re both planted. The trick is to choose a group of different plants whose total root space fits inside the bed, and whose overall harvest (each plant’s own yield plus any pairwise boosts) is as big as possible. The exact plant sizes, base yields and companion boosts are shown below.\n\n{\n  \"plant_count\": 16,\n  \"root_space_capacity\": 3459,\n  \"plant_ids\": [\n    0,\n    1,\n    2,\n    3,\n    4,\n    5,\n    6,\n    7,\n    8,\n    9,\n    10,\n    11,\n    12,\n    13,\n    14,\n    15\n  ],\n  \"linear\": [\n    {\n      \"plant_id\": 0,\n      \"base_yield\": 292\n    },\n    {\n      \"plant_id\": 1,\n      \"base_yield\": 154\n    },\n    {\n      \"plant_id\": 2,\n      \"base_yield\": 166\n    },\n    {\n      \"plant_id\": 3,\n      \"base_yield\": 240\n    },\n    {\n      \"plant_id\": 4,\n      \"base_yield\": 202\n    },\n    {\n      \"plant_id\": 5,\n      \"base_yield\": 144\n    },\n    {\n      \"plant_id\": 6,\n      \"base_yield\": 109\n    },\n    {\n      \"plant_id\": 7,\n      \"base_yield\": 219\n    },\n    {\n      \"plant_id\": 8,\n      \"base_yield\": 213\n    },\n    {\n      \"plant_id\": 9,\n      \"base_yield\": 116\n    },\n    {\n      \"plant_id\": 10,\n      \"base_yield\": 299\n    },\n    {\n      \"plant_id\": 11,\n      \"base_yield\": 126\n    },\n    {\n      \"plant_id\": 12,\n      \"base_yield\": 253\n    },\n    {\n      \"plant_id\": 13,\n      \"base_yield\": 164\n    },\n    {\n      \"plant_id\": 14,\n      \"base_yield\": 226\n    },\n    {\n      \"plant_id\": 15,\n      \"base_yield\": 151\n    }\n  ],\n  \"weights\": [\n    {\n      \"plant_id\": 0,\n      \"root_space\": 175\n    },\n    {\n      \"plant_id\": 1,\n      \"root_space\": 220\n    },\n    {\n      \"plant_id\": 2,\n      \"root_space\": 241\n    },\n    {\n      \"plant_id\": 3,\n      \"root_space\": 291\n    },\n    {\n      \"plant_id\": 4,\n      \"root_space\": 204\n    },\n    {\n      \"plant_id\": 5,\n      \"root_space\": 112\n    },\n    {\n      \"plant_id\": 6,\n      \"root_space\": 247\n    },\n    {\n      \"plant_id\": 7,\n      \"root_space\": 242\n    },\n    {\n      \"plant_id\": 8,\n      \"root_space\": 205\n    },\n    {\n      \"plant_id\": 9,\n      \"root_space\": 257\n    },\n    {\n      \"plant_id\": 10,\n      \"root_space\": 240\n    },\n    {\n      \"plant_id\": 11,\n      \"root_space\": 241\n    },\n    {\n      \"plant_id\": 12,\n      \"root_space\": 278\n    },\n    {\n      \"plant_id\": 13,\n      \"root_space\": 104\n    },\n    {\n      \"plant_id\": 14,\n      \"root_space\": 280\n    },\n    {\n      \"plant_id\": 15,\n      \"root_space\": 233\n    }\n  ],\n  \"quadratic\": [\n    {\n      \"plant_i_id\": 0,\n      \"plant_j_id\": 1,\n      \"companion_bonus\": 229\n    },\n    {\n      \"plant_i_id\": 0,\n      \"plant_j_id\": 2,\n      \"companion_bonus\": 278\n    },\n    {\n      \"plant_i_id\": 0,\n      \"plant_j_id\": 3,\n      \"companion_bonus\": 125\n    },\n    {\n      \"plant_i_id\": 0,\n      \"plant_j_id\": 4,\n      \"companion_bonus\": 147\n    },\n    {\n      \"plant_i_id\": 0,\n      \"plant_j_id\": 5,\n      \"companion_bonus\": 196\n    },\n    {\n      \"plant_i_id\": 0,\n      \"plant_j_id\": 6,\n      \"companion_bonus\": 222\n    },\n    {\n      \"plant_i_id\": 0,\n      \"plant_j_id\": 7,\n      \"companion_bonus\": 198\n    },\n    {\n      \"plant_i_id\": 0,\n      \"plant_j_id\": 8,\n      \"companion_bonus\": 282\n    },\n    {\n      \"plant_i_id\": 0,\n      \"plant_j_id\": 9,\n      \"companion_bonus\": 197\n    },\n    {\n      \"plant_i_id\": 0,\n      \"plant_j_id\": 10,\n      \"companion_bonus\": 123\n    },\n    {\n      \"plant_i_id\": 0,\n      \"plant_j_id\": 11,\n      \"companion_bonus\": 145\n    },\n    {\n      \"plant_i_id\": 0,\n      \"plant_j_id\": 12,\n      \"companion_bonus\": 255\n    },\n    {\n      \"plant_i_id\": 0,\n      \"plant_j_id\": 13,\n      \"companion_bonus\": 137\n    },\n    {\n      \"plant_i_id\": 0,\n      \"plant_j_id\": 14,\n      \"companion_bonus\": 107\n    },\n    {\n      \"plant_i_id\": 0,\n      \"plant_j_id\": 15,\n      \"companion_bonus\": 177\n    },\n    {\n      \"plant_i_id\": 1,\n      \"plant_j_id\": 2,\n      \"companion_bonus\": 247\n    },\n    {\n      \"plant_i_id\": 1,\n      \"plant_j_id\": 3,\n      \"companion_bonus\": 195\n    },\n    {\n      \"plant_i_id\": 1,\n      \"plant_j_id\": 4,\n      \"companion_bonus\": 175\n    },\n    {\n      \"plant_i_id\": 1,\n      \"plant_j_id\": 5,\n      \"companion_bonus\": 130\n    },\n    {\n      \"plant_i_id\": 1,\n      \"plant_j_id\": 6,\n      \"companion_bonus\": 172\n    },\n    {\n      \"plant_i_id\": 1,\n      \"plant_j_id\": 7,\n      \"companion_bonus\": 294\n    },\n    {\n      \"plant_i_id\": 1,\n      \"plant_j_id\": 8,\n      \"companion_bonus\": 227\n    },\n    {\n      \"plant_i_id\": 1,\n      \"plant_j_id\": 9,\n      \"companion_bonus\": 139\n    },\n    {\n      \"plant_i_id\": 1,\n      \"plant_j_id\": 10,\n      \"companion_bonus\": 130\n    },\n    {\n      \"plant_i_id\": 1,\n      \"plant_j_id\": 11,\n      \"companion_bonus\": 238\n    },\n    {\n      \"plant_i_id\": 1,\n      \"plant_j_id\": 12,\n      \"companion_bonus\": 249\n    },\n    {\n      \"plant_i_id\": 1,\n      \"plant_j_id\": 13,\n      \"companion_bonus\": 178\n    },\n    {\n      \"plant_i_id\": 1,\n      \"plant_j_id\": 14,\n      \"companion_bonus\": 252\n    },\n    {\n      \"plant_i_id\": 1,\n      \"plant_j_id\": 15,\n      \"companion_bonus\": 231\n    },\n    {\n      \"plant_i_id\": 2,\n      \"plant_j_id\": 3,\n      \"companion_bonus\": 214\n    },\n    {\n      \"plant_i_id\": 2,\n      \"plant_j_id\": 4,\n      \"companion_bonus\": 286\n    },\n    {\n      \"plant_i_id\": 2,\n      \"plant_j_id\": 5,\n      \"companion_bonus\": 211\n    },\n    {\n      \"plant_i_id\": 2,\n      \"plant_j_id\": 6,\n      \"companion_bonus\": 278\n    },\n    {\n      \"plant_i_id\": 2,\n      \"plant_j_id\": 7,\n      \"companion_bonus\": 212\n    },\n    {\n      \"plant_i_id\": 2,\n      \"plant_j_id\": 8,\n      \"companion_bonus\": 263\n    },\n    {\n      \"plant_i_id\": 2,\n      \"plant_j_id\": 9,\n      \"companion_bonus\": 156\n    },\n    {\n      \"plant_i_id\": 2,\n      \"plant_j_id\": 10,\n      \"companion_bonus\": 119\n    },\n    {\n      \"plant_i_id\": 2,\n      \"plant_j_id\": 11,\n      \"companion_bonus\": 297\n    },\n    {\n      \"plant_i_id\": 2,\n      \"plant_j_id\": 12,\n      \"companion_bonus\": 254\n    },\n    {\n      \"plant_i_id\": 2,\n      \"plant_j_id\": 13,\n      \"companion_bonus\": 234\n    },\n    {\n      \"plant_i_id\": 2,\n      \"plant_j_id\": 14,\n      \"companion_bonus\": 275\n    },\n    {\n      \"plant_i_id\": 2,\n      \"plant_j_id\": 15,\n      \"companion_bonus\": 207\n    },\n    {\n      \"plant_i_id\": 3,\n      \"plant_j_id\": 4,\n      \"companion_bonus\": 278\n    },\n    {\n      \"plant_i_id\": 3,\n      \"plant_j_id\": 5,\n      \"companion_bonus\": 150\n    },\n    {\n      \"plant_i_id\": 3,\n      \"plant_j_id\": 6,\n      \"companion_bonus\": 174\n    },\n    {\n      \"plant_i_id\": 3,\n      \"plant_j_id\": 7,\n      \"companion_bonus\": 126\n    },\n    {\n      \"plant_i_id\": 3,\n      \"plant_j_id\": 8,\n      \"companion_bonus\": 289\n    },\n    {\n      \"plant_i_id\": 3,\n      \"plant_j_id\": 9,\n      \"companion_bonus\": 246\n    },\n    {\n      \"plant_i_id\": 3,\n      \"plant_j_id\": 10,\n      \"companion_bonus\": 161\n    },\n    {\n      \"plant_i_id\": 3,\n      \"plant_j_id\": 11,\n      \"companion_bonus\": 222\n    },\n    {\n      \"plant_i_id\": 3,\n      \"plant_j_id\": 12,\n      \"companion_bonus\": 170\n    },\n    {\n      \"plant_i_id\": 3,\n      \"plant_j_id\": 13,\n      \"companion_bonus\": 101\n    },\n    {\n      \"plant_i_id\": 3,\n      \"plant_j_id\": 14,\n      \"companion_bonus\": 203\n    },\n    {\n      \"plant_i_id\": 3,\n      \"plant_j_id\": 15,\n      \"companion_bonus\": 170\n    },\n    {\n      \"plant_i_id\": 4,\n      \"plant_j_id\": 5,\n      \"companion_bonus\": 135\n    },\n    {\n      \"plant_i_id\": 4,\n      \"plant_j_id\": 6,\n      \"companion_bonus\": 173\n    },\n    {\n      \"plant_i_id\": 4,\n      \"plant_j_id\": 7,\n      \"companion_bonus\": 189\n    },\n    {\n      \"plant_i_id\": 4,\n      \"plant_j_id\": 8,\n      \"companion_bonus\": 103\n    },\n    {\n      \"plant_i_id\": 4,\n      \"plant_j_id\": 9,\n      \"companion_bonus\": 229\n    },\n    {\n      \"plant_i_id\": 4,\n      \"plant_j_id\": 10,\n      \"companion_bonus\": 209\n    },\n    {\n      \"plant_i_id\": 4,\n      \"plant_j_id\": 11,\n      \"companion_bonus\": 181\n    },\n    {\n      \"plant_i_id\": 4,\n      \"plant_j_id\": 12,\n      \"companion_bonus\": 140\n    },\n    {\n      \"plant_i_id\": 4,\n      \"plant_j_id\": 13,\n      \"companion_bonus\": 148\n    },\n    {\n      \"plant_i_id\": 4,\n      \"plant_j_id\": 14,\n      \"companion_bonus\": 220\n    },\n    {\n      \"plant_i_id\": 4,\n      \"plant_j_id\": 15,\n      \"companion_bonus\": 245\n    },\n    {\n      \"plant_i_id\": 5,\n      \"plant_j_id\": 6,\n      \"companion_bonus\": 191\n    },\n    {\n      \"plant_i_id\": 5,\n      \"plant_j_id\": 7,\n      \"companion_bonus\": 263\n    },\n    {\n      \"plant_i_id\": 5,\n      \"plant_j_id\": 8,\n      \"companion_bonus\": 104\n    },\n    {\n      \"plant_i_id\": 5,\n      \"plant_j_id\": 9,\n      \"companion_bonus\": 256\n    },\n    {\n      \"plant_i_id\": 5,\n      \"plant_j_id\": 10,\n      \"companion_bonus\": 198\n    },\n    {\n      \"plant_i_id\": 5,\n      \"plant_j_id\": 11,\n      \"companion_bonus\": 251\n    },\n    {\n      \"plant_i_id\": 5,\n      \"plant_j_id\": 12,\n      \"companion_bonus\": 122\n    },\n    {\n      \"plant_i_id\": 5,\n      \"plant_j_id\": 13,\n      \"companion_bonus\": 295\n    },\n    {\n      \"plant_i_id\": 5,\n      \"plant_j_id\": 14,\n      \"companion_bonus\": 290\n    },\n    {\n      \"plant_i_id\": 5,\n      \"plant_j_id\": 15,\n      \"companion_bonus\": 249\n    },\n    {\n      \"plant_i_id\": 6,\n      \"plant_j_id\": 7,\n      \"companion_bonus\": 228\n    },\n    {\n      \"plant_i_id\": 6,\n      \"plant_j_id\": 8,\n      \"companion_bonus\": 197\n    },\n    {\n      \"plant_i_id\": 6,\n      \"plant_j_id\": 9,\n      \"companion_bonus\": 132\n    },\n    {\n      \"plant_i_id\": 6,\n      \"plant_j_id\": 10,\n      \"companion_bonus\": 264\n    },\n    {\n      \"plant_i_id\": 6,\n      \"plant_j_id\": 11,\n      \"companion_bonus\": 291\n    },\n    {\n      \"plant_i_id\": 6,\n      \"plant_j_id\": 12,\n      \"companion_bonus\": 165\n    },\n    {\n      \"plant_i_id\": 6,\n      \"plant_j_id\": 13,\n      \"companion_bonus\": 127\n    },\n    {\n      \"plant_i_id\": 6,\n      \"plant_j_id\": 14,\n      \"companion_bonus\": 190\n    },\n    {\n      \"plant_i_id\": 6,\n      \"plant_j_id\": 15,\n      \"companion_bonus\": 126\n    },\n    {\n      \"plant_i_id\": 7,\n      \"plant_j_id\": 8,\n      \"companion_bonus\": 264\n    },\n    {\n      \"plant_i_id\": 7,\n      \"plant_j_id\": 9,\n      \"companion_bonus\": 146\n    },\n    {\n      \"plant_i_id\": 7,\n      \"plant_j_id\": 10,\n      \"companion_bonus\": 270\n    },\n    {\n      \"plant_i_id\": 7,\n      \"plant_j_id\": 11,\n      \"companion_bonus\": 201\n    },\n    {\n      \"plant_i_id\": 7,\n      \"plant_j_id\": 12,\n      \"companion_bonus\": 225\n    },\n    {\n      \"plant_i_id\": 7,\n      \"plant_j_id\": 13,\n      \"companion_bonus\": 114\n    },\n    {\n      \"plant_i_id\": 7,\n      \"plant_j_id\": 14,\n      \"companion_bonus\": 136\n    },\n    {\n      \"plant_i_id\": 7,\n      \"plant_j_id\": 15,\n      \"companion_bonus\": 139\n    },\n    {\n      \"plant_i_id\": 8,\n      \"plant_j_id\": 9,\n      \"companion_bonus\": 123\n    },\n    {\n      \"plant_i_id\": 8,\n      \"plant_j_id\": 10,\n      \"companion_bonus\": 198\n    },\n    {\n      \"plant_i_id\": 8,\n      \"plant_j_id\": 11,\n      \"companion_bonus\": 189\n    },\n    {\n      \"plant_i_id\": 8,\n      \"plant_j_id\": 12,\n      \"companion_bonus\": 242\n    },\n    {\n      \"plant_i_id\": 8,\n      \"plant_j_id\": 13,\n      \"companion_bonus\": 222\n    },\n    {\n      \"plant_i_id\": 8,\n      \"plant_j_id\": 14,\n      \"companion_bonus\": 151\n    },\n    {\n      \"plant_i_id\": 8,\n      \"plant_j_id\": 15,\n      \"companion_bonus\": 143\n    },\n    {\n      \"plant_i_id\": 9,\n      \"plant_j_id\": 10,\n      \"companion_bonus\": 247\n    },\n    {\n      \"plant_i_id\": 9,\n      \"plant_j_id\": 11,\n      \"companion_bonus\": 258\n    },\n    {\n      \"plant_i_id\": 9,\n      \"plant_j_id\": 12,\n      \"companion_bonus\": 298\n    },\n    {\n      \"plant_i_id\": 9,\n      \"plant_j_id\": 13,\n      \"companion_bonus\": 154\n    },\n    {\n      \"plant_i_id\": 9,\n      \"plant_j_id\": 14,\n      \"companion_bonus\": 142\n    },\n    {\n      \"plant_i_id\": 9,\n      \"plant_j_id\": 15,\n      \"companion_bonus\": 215\n    },\n    {\n      \"plant_i_id\": 10,\n      \"plant_j_id\": 11,\n      \"companion_bonus\": 185\n    },\n    {\n      \"plant_i_id\": 10,\n      \"plant_j_id\": 12,\n      \"companion_bonus\": 251\n    },\n    {\n      \"plant_i_id\": 10,\n      \"plant_j_id\": 13,\n      \"companion_bonus\": 203\n    },\n    {\n      \"plant_i_id\": 10,\n      \"plant_j_id\": 14,\n      \"companion_bonus\": 114\n    },\n    {\n      \"plant_i_id\": 10,\n      \"plant_j_id\": 15,\n      \"companion_bonus\": 126\n    },\n    {\n      \"plant_i_id\": 11,\n      \"plant_j_id\": 12,\n      \"companion_bonus\": 118\n    },\n    {\n      \"plant_i_id\": 11,\n      \"plant_j_id\": 13,\n      \"companion_bonus\": 218\n    },\n    {\n      \"plant_i_id\": 11,\n      \"plant_j_id\": 14,\n      \"companion_bonus\": 219\n    },\n    {\n      \"plant_i_id\": 11,\n      \"plant_j_id\": 15,\n      \"companion_bonus\": 273\n    },\n    {\n      \"plant_i_id\": 12,\n      \"plant_j_id\": 13,\n      \"companion_bonus\": 273\n    },\n    {\n      \"plant_i_id\": 12,\n      \"plant_j_id\": 14,\n      \"companion_bonus\": 272\n    },\n    {\n      \"plant_i_id\": 12,\n      \"plant_j_id\": 15,\n      \"companion_bonus\": 268\n    },\n    {\n      \"plant_i_id\": 13,\n      \"plant_j_id\": 14,\n      \"companion_bonus\": 205\n    },\n    {\n      \"plant_i_id\": 13,\n      \"plant_j_id\": 15,\n      \"companion_bonus\": 241\n    },\n    {\n      \"plant_i_id\": 14,\n      \"plant_j_id\": 15,\n      \"companion_bonus\": 118\n    }\n  ]\n}\n\nOh, and when you're ready to tell me which plants to plant, just pop your pick into this tiny JSON layout so it's straightforward to read:\n\n{\n  \"solution\": [<selected_item_id>, <selected_item_id>, ...]\n}\n\nPretty simple: \"solution\" is just a list of the plant IDs you want in the bed. Think of it like a short shopping list — put each plant's identifier in the array, nothing else. This JSON is just a sketch of the shape I expect, not the actual answer itself.\n\nPlease make sure to use the identifiers exactly as they appear in the instance input — no renaming and no new labels. For example: \"Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.\"",
    "instance": {
      "linear_coeffs": [
        292,
        154,
        166,
        240,
        202,
        144,
        109,
        219,
        213,
        116,
        299,
        126,
        253,
        164,
        226,
        151
      ],
      "quadratic_coeffs": [
        [
          226,
          229,
          278,
          125,
          147,
          196,
          222,
          198,
          282,
          197,
          123,
          145,
          255,
          137,
          107,
          177
        ],
        [
          0,
          273,
          247,
          195,
          175,
          130,
          172,
          294,
          227,
          139,
          130,
          238,
          249,
          178,
          252,
          231
        ],
        [
          0,
          0,
          190,
          214,
          286,
          211,
          278,
          212,
          263,
          156,
          119,
          297,
          254,
          234,
          275,
          207
        ],
        [
          0,
          0,
          0,
          255,
          278,
          150,
          174,
          126,
          289,
          246,
          161,
          222,
          170,
          101,
          203,
          170
        ],
        [
          0,
          0,
          0,
          0,
          288,
          135,
          173,
          189,
          103,
          229,
          209,
          181,
          140,
          148,
          220,
          245
        ],
        [
          0,
          0,
          0,
          0,
          0,
          261,
          191,
          263,
          104,
          256,
          198,
          251,
          122,
          295,
          290,
          249
        ],
        [
          0,
          0,
          0,
          0,
          0,
          0,
          222,
          228,
          197,
          132,
          264,
          291,
          165,
          127,
          190,
          126
        ],
        [
          0,
          0,
          0,
          0,
          0,
          0,
          0,
          176,
          264,
          146,
          270,
          201,
          225,
          114,
          136,
          139
        ],
        [
          0,
          0,
          0,
          0,
          0,
          0,
          0,
          0,
          212,
          123,
          198,
          189,
          242,
          222,
          151,
          143
        ],
        [
          0,
          0,
          0,
          0,
          0,
          0,
          0,
          0,
          0,
          133,
          247,
          258,
          298,
          154,
          142,
          215
        ],
        [
          0,
          0,
          0,
          0,
          0,
          0,
          0,
          0,
          0,
          0,
          131,
          185,
          251,
          203,
          114,
          126
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    "instruction": "Recently the team had to prioritize features for a tight release, and the conversation boiled down to picking a set that actually fits the sprint. Each feature takes a chunk of time and delivers a baseline user value; on top of that, certain feature pairs multiply the perceived value when they’re both delivered. The trick is to assemble features so the total time stays inside the sprint budget and the total user value equals the sum of chosen features’ base values plus any extra pair bonuses for combinations. Nothing can be split or chosen twice, and the precise numbers are shown below.\n\n{\n  \"num_features\": 18,\n  \"sprint_effort_budget\": 2297,\n  \"feature_ids\": [\n    \"A\",\n    \"B\",\n    \"C\",\n    \"D\",\n    \"E\",\n    \"F\",\n    \"G\",\n    \"H\",\n    \"I\",\n    \"J\",\n    \"K\",\n    \"L\",\n    \"M\",\n    \"N\",\n    \"O\",\n    \"P\",\n    \"Q\",\n    \"R\"\n  ],\n  \"linear\": [\n    {\n      \"feature_id\": \"A\",\n      \"base_user_value\": 412\n    },\n    {\n      \"feature_id\": \"B\",\n      \"base_user_value\": 431\n    },\n    {\n      \"feature_id\": \"C\",\n      \"base_user_value\": 580\n    },\n    {\n      \"feature_id\": \"D\",\n      \"base_user_value\": 900\n    },\n    {\n      \"feature_id\": \"E\",\n      \"base_user_value\": 492\n    },\n    {\n      \"feature_id\": \"F\",\n      \"base_user_value\": 852\n    },\n    {\n      \"feature_id\": \"G\",\n      \"base_user_value\": 902\n    },\n    {\n      \"feature_id\": \"H\",\n      \"base_user_value\": 537\n    },\n    {\n      \"feature_id\": \"I\",\n      \"base_user_value\": 739\n    },\n    {\n      \"feature_id\": \"J\",\n      \"base_user_value\": 558\n    },\n    {\n      \"feature_id\": \"K\",\n      \"base_user_value\": 356\n    },\n    {\n      \"feature_id\": \"L\",\n      \"base_user_value\": 219\n    },\n    {\n      \"feature_id\": \"M\",\n      \"base_user_value\": 780\n    },\n    {\n      \"feature_id\": \"N\",\n      \"base_user_value\": 146\n    },\n    {\n      \"feature_id\": \"O\",\n      \"base_user_value\": 634\n    },\n    {\n      \"feature_id\": \"P\",\n      \"base_user_value\": 262\n    },\n    {\n      \"feature_id\": \"Q\",\n      \"base_user_value\": 826\n    },\n    {\n      \"feature_id\": \"R\",\n      \"base_user_value\": 632\n    }\n  ],\n  \"weights\": [\n    {\n      \"feature_id\": \"A\",\n      \"development_effort\": 415\n    },\n    {\n      \"feature_id\": \"B\",\n      \"development_effort\": 397\n    },\n    {\n      \"feature_id\": \"C\",\n      \"development_effort\": 590\n    },\n    {\n      \"feature_id\": \"D\",\n      \"development_effort\": 887\n    },\n    {\n      \"feature_id\": \"E\",\n      \"development_effort\": 455\n    },\n    {\n      \"feature_id\": \"F\",\n      \"development_effort\": 837\n    },\n    {\n      \"feature_id\": \"G\",\n      \"development_effort\": 899\n    },\n    {\n      \"feature_id\": \"H\",\n      \"development_effort\": 575\n    },\n    {\n      \"feature_id\": \"I\",\n      \"development_effort\": 726\n    },\n    {\n      \"feature_id\": \"J\",\n      \"development_effort\": 555\n    },\n    {\n      \"feature_id\": \"K\",\n      \"development_effort\": 359\n    },\n    {\n      \"feature_id\": \"L\",\n      \"development_effort\": 255\n    },\n    {\n      \"feature_id\": \"M\",\n      \"development_effort\": 774\n    },\n    {\n      \"feature_id\": \"N\",\n      \"development_effort\": 118\n    },\n    {\n      \"feature_id\": \"O\",\n      \"development_effort\": 661\n    },\n    {\n      \"feature_id\": \"P\",\n      \"development_effort\": 304\n    },\n    {\n      \"feature_id\": \"Q\",\n      \"development_effort\": 853\n    },\n    {\n      \"feature_id\": \"R\",\n      \"development_effort\": 671\n    }\n  ],\n  \"quadratic\": [\n    {\n      \"feature_i_id\": \"A\",\n      \"feature_j_id\": \"B\",\n      \"interaction_value\": 605\n    },\n    {\n      \"feature_i_id\": \"A\",\n      \"feature_j_id\": \"C\",\n      \"interaction_value\": 553\n    },\n    {\n      \"feature_i_id\": \"A\",\n      \"feature_j_id\": \"D\",\n      \"interaction_value\": 559\n    },\n    {\n      \"feature_i_id\": \"A\",\n      \"feature_j_id\": \"E\",\n      \"interaction_value\": 255\n    },\n    {\n      \"feature_i_id\": \"A\",\n      \"feature_j_id\": \"F\",\n      \"interaction_value\": 275\n    },\n    {\n      \"feature_i_id\": \"A\",\n      \"feature_j_id\": \"G\",\n      \"interaction_value\": 463\n    },\n    {\n      \"feature_i_id\": \"A\",\n      \"feature_j_id\": \"H\",\n      \"interaction_value\": 448\n    },\n    {\n      \"feature_i_id\": \"A\",\n      \"feature_j_id\": \"I\",\n      \"interaction_value\": 425\n    },\n    {\n      \"feature_i_id\": \"A\",\n      \"feature_j_id\": \"J\",\n      \"interaction_value\": 401\n    },\n    {\n      \"feature_i_id\": \"A\",\n      \"feature_j_id\": \"K\",\n      \"interaction_value\": 603\n    },\n    {\n      \"feature_i_id\": \"A\",\n      \"feature_j_id\": \"L\",\n      \"interaction_value\": 604\n    },\n    {\n      \"feature_i_id\": \"A\",\n      \"feature_j_id\": \"M\",\n      \"interaction_value\": 269\n    },\n    {\n      \"feature_i_id\": \"A\",\n      \"feature_j_id\": \"N\",\n      \"interaction_value\": 594\n    },\n    {\n      \"feature_i_id\": \"A\",\n      \"feature_j_id\": \"O\",\n      \"interaction_value\": 307\n    },\n    {\n      \"feature_i_id\": \"A\",\n      \"feature_j_id\": \"P\",\n      \"interaction_value\": 503\n    },\n    {\n      \"feature_i_id\": \"A\",\n      \"feature_j_id\": \"Q\",\n      \"interaction_value\": 433\n    },\n    {\n      \"feature_i_id\": \"A\",\n      \"feature_j_id\": \"R\",\n      \"interaction_value\": 438\n    },\n    {\n      \"feature_i_id\": \"B\",\n      \"feature_j_id\": \"C\",\n      \"interaction_value\": 355\n    },\n    {\n      \"feature_i_id\": \"B\",\n      \"feature_j_id\": \"D\",\n      \"interaction_value\": 415\n    },\n    {\n      \"feature_i_id\": \"B\",\n      \"feature_j_id\": \"E\",\n      \"interaction_value\": 269\n    },\n    {\n      \"feature_i_id\": \"B\",\n      \"feature_j_id\": \"F\",\n      \"interaction_value\": 477\n    },\n    {\n      \"feature_i_id\": \"B\",\n      \"feature_j_id\": \"G\",\n      \"interaction_value\": 489\n    },\n    {\n      \"feature_i_id\": \"B\",\n      \"feature_j_id\": \"H\",\n      \"interaction_value\": 321\n    },\n    {\n      \"feature_i_id\": \"B\",\n      \"feature_j_id\": \"I\",\n      \"interaction_value\": 469\n    },\n    {\n      \"feature_i_id\": \"B\",\n      \"feature_j_id\": \"J\",\n      \"interaction_value\": 253\n    },\n    {\n      \"feature_i_id\": \"B\",\n      \"feature_j_id\": \"K\",\n      \"interaction_value\": 591\n    },\n    {\n      \"feature_i_id\": \"B\",\n      \"feature_j_id\": \"L\",\n      \"interaction_value\": 504\n    },\n    {\n      \"feature_i_id\": \"B\",\n      \"feature_j_id\": \"M\",\n      \"interaction_value\": 381\n    },\n    {\n      \"feature_i_id\": \"B\",\n      \"feature_j_id\": \"N\",\n      \"interaction_value\": 492\n    },\n    {\n      \"feature_i_id\": \"B\",\n      \"feature_j_id\": \"O\",\n      \"interaction_value\": 347\n    },\n    {\n      \"feature_i_id\": \"B\",\n      \"feature_j_id\": \"P\",\n      \"interaction_value\": 477\n    },\n    {\n      \"feature_i_id\": \"B\",\n      \"feature_j_id\": \"Q\",\n      \"interaction_value\": 537\n    },\n    {\n      \"feature_i_id\": \"B\",\n      \"feature_j_id\": \"R\",\n      \"interaction_value\": 453\n    },\n    {\n      \"feature_i_id\": \"C\",\n      \"feature_j_id\": \"D\",\n      \"interaction_value\": 681\n    },\n    {\n      \"feature_i_id\": \"C\",\n      \"feature_j_id\": \"E\",\n      \"interaction_value\": 572\n    },\n    {\n      \"feature_i_id\": \"C\",\n      \"feature_j_id\": \"F\",\n      \"interaction_value\": 640\n    },\n    {\n      \"feature_i_id\": \"C\",\n      \"feature_j_id\": \"G\",\n      \"interaction_value\": 453\n    },\n    {\n      \"feature_i_id\": \"C\",\n      \"feature_j_id\": \"H\",\n      \"interaction_value\": 684\n    },\n    {\n      \"feature_i_id\": \"C\",\n      \"feature_j_id\": \"I\",\n      \"interaction_value\": 400\n    },\n    {\n      \"feature_i_id\": \"C\",\n      \"feature_j_id\": \"J\",\n      \"interaction_value\": 268\n    },\n    {\n      \"feature_i_id\": \"C\",\n      \"feature_j_id\": \"K\",\n      \"interaction_value\": 387\n    },\n    {\n      \"feature_i_id\": \"C\",\n      \"feature_j_id\": \"L\",\n      \"interaction_value\": 553\n    },\n    {\n      \"feature_i_id\": \"C\",\n      \"feature_j_id\": \"M\",\n      \"interaction_value\": 301\n    },\n    {\n      \"feature_i_id\": \"C\",\n      \"feature_j_id\": \"N\",\n      \"interaction_value\": 718\n    },\n    {\n      \"feature_i_id\": \"C\",\n      \"feature_j_id\": \"O\",\n      \"interaction_value\": 710\n    },\n    {\n      \"feature_i_id\": \"C\",\n      \"feature_j_id\": \"P\",\n      \"interaction_value\": 453\n    },\n    {\n      \"feature_i_id\": \"C\",\n      \"feature_j_id\": \"Q\",\n      \"interaction_value\": 458\n    },\n    {\n      \"feature_i_id\": \"C\",\n      \"feature_j_id\": \"R\",\n      \"interaction_value\": 657\n    },\n    {\n      \"feature_i_id\": \"D\",\n      \"feature_j_id\": \"E\",\n      \"interaction_value\": 770\n    },\n    {\n      \"feature_i_id\": \"D\",\n      \"feature_j_id\": \"F\",\n      \"interaction_value\": 330\n    },\n    {\n      \"feature_i_id\": \"D\",\n      \"feature_j_id\": \"G\",\n      \"interaction_value\": 839\n    },\n    {\n      \"feature_i_id\": \"D\",\n      \"feature_j_id\": \"H\",\n      \"interaction_value\": 300\n    },\n    {\n      \"feature_i_id\": \"D\",\n      \"feature_j_id\": \"I\",\n      \"interaction_value\": 836\n    },\n    {\n      \"feature_i_id\": \"D\",\n      \"feature_j_id\": \"J\",\n      \"interaction_value\": 376\n    },\n    {\n      \"feature_i_id\": \"D\",\n      \"feature_j_id\": \"K\",\n      \"interaction_value\": 330\n    },\n    {\n      \"feature_i_id\": \"D\",\n      \"feature_j_id\": \"L\",\n      \"interaction_value\": 537\n    },\n    {\n      \"feature_i_id\": \"D\",\n      \"feature_j_id\": \"M\",\n      \"interaction_value\": 339\n    },\n    {\n      \"feature_i_id\": \"D\",\n      \"feature_j_id\": \"N\",\n      \"interaction_value\": 686\n    },\n    {\n      \"feature_i_id\": \"D\",\n      \"feature_j_id\": \"O\",\n      \"interaction_value\": 794\n    },\n    {\n      \"feature_i_id\": \"D\",\n      \"feature_j_id\": \"P\",\n      \"interaction_value\": 755\n    },\n    {\n      \"feature_i_id\": \"D\",\n      \"feature_j_id\": \"Q\",\n      \"interaction_value\": 382\n    },\n    {\n      \"feature_i_id\": \"D\",\n      \"feature_j_id\": \"R\",\n      \"interaction_value\": 619\n    },\n    {\n      \"feature_i_id\": \"E\",\n      \"feature_j_id\": \"F\",\n      \"interaction_value\": 264\n    },\n    {\n      \"feature_i_id\": \"E\",\n      \"feature_j_id\": \"G\",\n      \"interaction_value\": 516\n    },\n    {\n      \"feature_i_id\": \"E\",\n      \"feature_j_id\": \"H\",\n      \"interaction_value\": 567\n    },\n    {\n      \"feature_i_id\": \"E\",\n      \"feature_j_id\": \"I\",\n      \"interaction_value\": 548\n    },\n    {\n      \"feature_i_id\": \"E\",\n      \"feature_j_id\": \"J\",\n      \"interaction_value\": 371\n    },\n    {\n      \"feature_i_id\": \"E\",\n      \"feature_j_id\": \"K\",\n      \"interaction_value\": 605\n    },\n    {\n      \"feature_i_id\": \"E\",\n      \"feature_j_id\": \"L\",\n      \"interaction_value\": 469\n    },\n    {\n      \"feature_i_id\": \"E\",\n      \"feature_j_id\": \"M\",\n      \"interaction_value\": 328\n    },\n    {\n      \"feature_i_id\": \"E\",\n      \"feature_j_id\": \"N\",\n      \"interaction_value\": 230\n    },\n    {\n      \"feature_i_id\": \"E\",\n      \"feature_j_id\": \"O\",\n      \"interaction_value\": 330\n    },\n    {\n      \"feature_i_id\": \"E\",\n      \"feature_j_id\": \"P\",\n      \"interaction_value\": 352\n    },\n    {\n      \"feature_i_id\": \"E\",\n      \"feature_j_id\": \"Q\",\n      \"interaction_value\": 344\n    },\n    {\n      \"feature_i_id\": \"E\",\n      \"feature_j_id\": \"R\",\n      \"interaction_value\": 339\n    },\n    {\n      \"feature_i_id\": \"F\",\n      \"feature_j_id\": \"G\",\n      \"interaction_value\": 436\n    },\n    {\n      \"feature_i_id\": \"F\",\n      \"feature_j_id\": \"H\",\n      \"interaction_value\": 399\n    },\n    {\n      \"feature_i_id\": \"F\",\n      \"feature_j_id\": \"I\",\n      \"interaction_value\": 701\n    },\n    {\n      \"feature_i_id\": \"F\",\n      \"feature_j_id\": \"J\",\n      \"interaction_value\": 445\n    },\n    {\n      \"feature_i_id\": \"F\",\n      \"feature_j_id\": \"K\",\n      \"interaction_value\": 637\n    },\n    {\n      \"feature_i_id\": \"F\",\n      \"feature_j_id\": \"L\",\n      \"interaction_value\": 597\n    },\n    {\n      \"feature_i_id\": \"F\",\n      \"feature_j_id\": \"M\",\n      \"interaction_value\": 248\n    },\n    {\n      \"feature_i_id\": \"F\",\n      \"feature_j_id\": \"N\",\n      \"interaction_value\": 218\n    },\n    {\n      \"feature_i_id\": \"F\",\n      \"feature_j_id\": \"O\",\n      \"interaction_value\": 283\n    },\n    {\n      \"feature_i_id\": \"F\",\n      \"feature_j_id\": \"P\",\n      \"interaction_value\": 395\n    },\n    {\n      \"feature_i_id\": \"F\",\n      \"feature_j_id\": \"Q\",\n      \"interaction_value\": 158\n    },\n    {\n      \"feature_i_id\": \"F\",\n      \"feature_j_id\": \"R\",\n      \"interaction_value\": 215\n    },\n    {\n      \"feature_i_id\": \"G\",\n      \"feature_j_id\": \"H\",\n      \"interaction_value\": 741\n    },\n    {\n      \"feature_i_id\": \"G\",\n      \"feature_j_id\": \"I\",\n      \"interaction_value\": 541\n    },\n    {\n      \"feature_i_id\": \"G\",\n      \"feature_j_id\": \"J\",\n      \"interaction_value\": 788\n    },\n    {\n      \"feature_i_id\": \"G\",\n      \"feature_j_id\": \"K\",\n      \"interaction_value\": 537\n    },\n    {\n      \"feature_i_id\": \"G\",\n      \"feature_j_id\": \"L\",\n      \"interaction_value\": 449\n    },\n    {\n      \"feature_i_id\": \"G\",\n      \"feature_j_id\": \"M\",\n      \"interaction_value\": 558\n    },\n    {\n      \"feature_i_id\": \"G\",\n      \"feature_j_id\": \"N\",\n      \"interaction_value\": 289\n    },\n    {\n      \"feature_i_id\": \"G\",\n      \"feature_j_id\": \"O\",\n      \"interaction_value\": 670\n    },\n    {\n      \"feature_i_id\": \"G\",\n      \"feature_j_id\": \"P\",\n      \"interaction_value\": 617\n    },\n    {\n      \"feature_i_id\": \"G\",\n      \"feature_j_id\": \"Q\",\n      \"interaction_value\": 495\n    },\n    {\n      \"feature_i_id\": \"G\",\n      \"feature_j_id\": \"R\",\n      \"interaction_value\": 452\n    },\n    {\n      \"feature_i_id\": \"H\",\n      \"feature_j_id\": \"I\",\n      \"interaction_value\": 369\n    },\n    {\n      \"feature_i_id\": \"H\",\n      \"feature_j_id\": \"J\",\n      \"interaction_value\": 273\n    },\n    {\n      \"feature_i_id\": \"H\",\n      \"feature_j_id\": \"K\",\n      \"interaction_value\": 656\n    },\n    {\n      \"feature_i_id\": \"H\",\n      \"feature_j_id\": \"L\",\n      \"interaction_value\": 319\n    },\n    {\n      \"feature_i_id\": \"H\",\n      \"feature_j_id\": \"M\",\n      \"interaction_value\": 301\n    },\n    {\n      \"feature_i_id\": \"H\",\n      \"feature_j_id\": \"N\",\n      \"interaction_value\": 411\n    },\n    {\n      \"feature_i_id\": \"H\",\n      \"feature_j_id\": \"O\",\n      \"interaction_value\": 524\n    },\n    {\n      \"feature_i_id\": \"H\",\n      \"feature_j_id\": \"P\",\n      \"interaction_value\": 659\n    },\n    {\n      \"feature_i_id\": \"H\",\n      \"feature_j_id\": \"Q\",\n      \"interaction_value\": 657\n    },\n    {\n      \"feature_i_id\": \"H\",\n      \"feature_j_id\": \"R\",\n      \"interaction_value\": 568\n    },\n    {\n      \"feature_i_id\": \"I\",\n      \"feature_j_id\": \"J\",\n      \"interaction_value\": 787\n    },\n    {\n      \"feature_i_id\": \"I\",\n      \"feature_j_id\": \"K\",\n      \"interaction_value\": 348\n    },\n    {\n      \"feature_i_id\": \"I\",\n      \"feature_j_id\": \"L\",\n      \"interaction_value\": 371\n    },\n    {\n      \"feature_i_id\": \"I\",\n      \"feature_j_id\": \"M\",\n      \"interaction_value\": 285\n    },\n    {\n      \"feature_i_id\": \"I\",\n      \"feature_j_id\": \"N\",\n      \"interaction_value\": 490\n    },\n    {\n      \"feature_i_id\": \"I\",\n      \"feature_j_id\": \"O\",\n      \"interaction_value\": 606\n    },\n    {\n      \"feature_i_id\": \"I\",\n      \"feature_j_id\": \"P\",\n      \"interaction_value\": 581\n    },\n    {\n      \"feature_i_id\": \"I\",\n      \"feature_j_id\": \"Q\",\n      \"interaction_value\": 319\n    },\n    {\n      \"feature_i_id\": \"I\",\n      \"feature_j_id\": \"R\",\n      \"interaction_value\": 506\n    },\n    {\n      \"feature_i_id\": \"J\",\n      \"feature_j_id\": \"K\",\n      \"interaction_value\": 589\n    },\n    {\n      \"feature_i_id\": \"J\",\n      \"feature_j_id\": \"L\",\n      \"interaction_value\": 212\n    },\n    {\n      \"feature_i_id\": \"J\",\n      \"feature_j_id\": \"M\",\n      \"interaction_value\": 381\n    },\n    {\n      \"feature_i_id\": \"J\",\n      \"feature_j_id\": \"N\",\n      \"interaction_value\": 435\n    },\n    {\n      \"feature_i_id\": \"J\",\n      \"feature_j_id\": \"O\",\n      \"interaction_value\": 378\n    },\n    {\n      \"feature_i_id\": \"J\",\n      \"feature_j_id\": \"P\",\n      \"interaction_value\": 231\n    },\n    {\n      \"feature_i_id\": \"J\",\n      \"feature_j_id\": \"Q\",\n      \"interaction_value\": 559\n    },\n    {\n      \"feature_i_id\": \"J\",\n      \"feature_j_id\": \"R\",\n      \"interaction_value\": 247\n    },\n    {\n      \"feature_i_id\": \"K\",\n      \"feature_j_id\": \"L\",\n      \"interaction_value\": 535\n    },\n    {\n      \"feature_i_id\": \"K\",\n      \"feature_j_id\": \"M\",\n      \"interaction_value\": 337\n    },\n    {\n      \"feature_i_id\": \"K\",\n      \"feature_j_id\": \"N\",\n      \"interaction_value\": 683\n    },\n    {\n      \"feature_i_id\": \"K\",\n      \"feature_j_id\": \"O\",\n      \"interaction_value\": 790\n    },\n    {\n      \"feature_i_id\": \"K\",\n      \"feature_j_id\": \"P\",\n      \"interaction_value\": 751\n    },\n    {\n      \"feature_i_id\": \"K\",\n      \"feature_j_id\": \"Q\",\n      \"interaction_value\": 561\n    },\n    {\n      \"feature_i_id\": \"K\",\n      \"feature_j_id\": \"R\",\n      \"interaction_value\": 512\n    },\n    {\n      \"feature_i_id\": \"L\",\n      \"feature_j_id\": \"M\",\n      \"interaction_value\": 684\n    },\n    {\n      \"feature_i_id\": \"L\",\n      \"feature_j_id\": \"N\",\n      \"interaction_value\": 585\n    },\n    {\n      \"feature_i_id\": \"L\",\n      \"feature_j_id\": \"O\",\n      \"interaction_value\": 416\n    },\n    {\n      \"feature_i_id\": \"L\",\n      \"feature_j_id\": \"P\",\n      \"interaction_value\": 383\n    },\n    {\n      \"feature_i_id\": \"L\",\n      \"feature_j_id\": \"Q\",\n      \"interaction_value\": 325\n    },\n    {\n      \"feature_i_id\": \"L\",\n      \"feature_j_id\": \"R\",\n      \"interaction_value\": 362\n    },\n    {\n      \"feature_i_id\": \"M\",\n      \"feature_j_id\": \"N\",\n      \"interaction_value\": 291\n    },\n    {\n      \"feature_i_id\": \"M\",\n      \"feature_j_id\": \"O\",\n      \"interaction_value\": 779\n    },\n    {\n      \"feature_i_id\": \"M\",\n      \"feature_j_id\": \"P\",\n      \"interaction_value\": 647\n    },\n    {\n      \"feature_i_id\": \"M\",\n      \"feature_j_id\": \"Q\",\n      \"interaction_value\": 770\n    },\n    {\n      \"feature_i_id\": \"M\",\n      \"feature_j_id\": \"R\",\n      \"interaction_value\": 487\n    },\n    {\n      \"feature_i_id\": \"N\",\n      \"feature_j_id\": \"O\",\n      \"interaction_value\": 608\n    },\n    {\n      \"feature_i_id\": \"N\",\n      \"feature_j_id\": \"P\",\n      \"interaction_value\": 267\n    },\n    {\n      \"feature_i_id\": \"N\",\n      \"feature_j_id\": \"Q\",\n      \"interaction_value\": 608\n    },\n    {\n      \"feature_i_id\": \"N\",\n      \"feature_j_id\": \"R\",\n      \"interaction_value\": 421\n    },\n    {\n      \"feature_i_id\": \"O\",\n      \"feature_j_id\": \"P\",\n      \"interaction_value\": 173\n    },\n    {\n      \"feature_i_id\": \"O\",\n      \"feature_j_id\": \"Q\",\n      \"interaction_value\": 402\n    },\n    {\n      \"feature_i_id\": \"O\",\n      \"feature_j_id\": \"R\",\n      \"interaction_value\": 392\n    },\n    {\n      \"feature_i_id\": \"P\",\n      \"feature_j_id\": \"Q\",\n      \"interaction_value\": 299\n    },\n    {\n      \"feature_i_id\": \"P\",\n      \"feature_j_id\": \"R\",\n      \"interaction_value\": 212\n    },\n    {\n      \"feature_i_id\": \"Q\",\n      \"feature_j_id\": \"R\",\n      \"interaction_value\": 330\n    }\n  ]\n}\n\nOh, and when you send back the chosen set, a tiny JSON snippet like this works great:\n\n{\n  \"solution\": [<selected_feature_id>, <selected_feature_id>, ...]\n}\n\nThink of \"solution\" as just a list of the features you want in the sprint — put each feature's identifier in that array, that's all. This JSON is just the shape I expect, not the final picks.\n\nPlease use the identifiers exactly as they appear in the instance input — do not rename them or invent new labels. For example: \"Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.\"",
    "instance": {
      "linear_coeffs": [
        412,
        431,
        580,
        900,
        492,
        852,
        902,
        537,
        739,
        558,
        356,
        219,
        780,
        146,
        634,
        262,
        826,
        632
      ],
      "quadratic_coeffs": [
        [
          543,
          605,
          553,
          559,
          255,
          275,
          463,
          448,
          425,
          401,
          603,
          604,
          269,
          594,
          307,
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          "quadratic_profit": 770
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        },
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        },
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          "item_i_id": "N",
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        },
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          "item_i_id": "N",
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          "quadratic_profit": 421
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          "item_i_id": "O",
          "item_j_id": "O",
          "quadratic_profit": 393
        },
        {
          "item_i_id": "O",
          "item_j_id": "P",
          "quadratic_profit": 173
        },
        {
          "item_i_id": "O",
          "item_j_id": "Q",
          "quadratic_profit": 402
        },
        {
          "item_i_id": "O",
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          "quadratic_profit": 392
        },
        {
          "item_i_id": "P",
          "item_j_id": "P",
          "quadratic_profit": 232
        },
        {
          "item_i_id": "P",
          "item_j_id": "Q",
          "quadratic_profit": 299
        },
        {
          "item_i_id": "P",
          "item_j_id": "R",
          "quadratic_profit": 212
        },
        {
          "item_i_id": "Q",
          "item_j_id": "Q",
          "quadratic_profit": 546
        },
        {
          "item_i_id": "Q",
          "item_j_id": "R",
          "quadratic_profit": 330
        }
      ]
    },
    "solution_variant": [
      "A",
      "B",
      "C",
      "K",
      "L",
      "N"
    ],
    "context_index": 7,
    "input_format": "json",
    "input_index_base": "names"
  },
  {
    "task_name": "QKP",
    "problem_type": "QKP",
    "instruction": "Many people don’t realize how much thought goes into a tasting menu: each plate has its own pull, certain combinations amplify that pull, and the kitchen can only handle a certain amount of prep and plating. With no dish duplicated, the challenge is to decide which courses to include so the guests’ total delight — the sum of every dish’s base delight plus any extra boost for pairs that appear together — is maximized without exceeding the kitchen’s capacity. The concrete course list, prep loads, and pairwise boosts are provided below.\n\n{\n  \"num_courses\": 22,\n  \"kitchen_capacity\": 1096,\n  \"course_ids\": [\n    \"A\",\n    \"B\",\n    \"C\",\n    \"D\",\n    \"E\",\n    \"F\",\n    \"G\",\n    \"H\",\n    \"I\",\n    \"J\",\n    \"K\",\n    \"L\",\n    \"M\",\n    \"N\",\n    \"O\",\n    \"P\",\n    \"Q\",\n    \"R\",\n    \"S\",\n    \"T\",\n    \"U\",\n    \"V\"\n  ],\n  \"linear\": [\n    {\n      \"course_id\": \"A\",\n      \"base_delight\": 126\n    },\n    {\n      \"course_id\": \"B\",\n      \"base_delight\": 116\n    },\n    {\n      \"course_id\": \"C\",\n      \"base_delight\": 140\n    },\n    {\n      \"course_id\": \"D\",\n      \"base_delight\": 152\n    },\n    {\n      \"course_id\": \"E\",\n      \"base_delight\": 103\n    },\n    {\n      \"course_id\": \"F\",\n      \"base_delight\": 117\n    },\n    {\n      \"course_id\": \"G\",\n      \"base_delight\": 119\n    },\n    {\n      \"course_id\": \"H\",\n      \"base_delight\": 121\n    },\n    {\n      \"course_id\": \"I\",\n      \"base_delight\": 124\n    },\n    {\n      \"course_id\": \"J\",\n      \"base_delight\": 112\n    },\n    {\n      \"course_id\": \"K\",\n      \"base_delight\": 142\n    },\n    {\n      \"course_id\": \"L\",\n      \"base_delight\": 138\n    },\n    {\n      \"course_id\": \"M\",\n      \"base_delight\": 133\n    },\n    {\n      \"course_id\": \"N\",\n      \"base_delight\": 120\n    },\n    {\n      \"course_id\": \"O\",\n      \"base_delight\": 119\n    },\n    {\n      \"course_id\": \"P\",\n      \"base_delight\": 111\n    },\n    {\n      \"course_id\": \"Q\",\n      \"base_delight\": 116\n    },\n    {\n      \"course_id\": \"R\",\n      \"base_delight\": 119\n    },\n    {\n      \"course_id\": \"S\",\n      \"base_delight\": 148\n    },\n    {\n      \"course_id\": \"T\",\n      \"base_delight\": 106\n    },\n    {\n      \"course_id\": \"U\",\n      \"base_delight\": 138\n    },\n    {\n      \"course_id\": \"V\",\n      \"base_delight\": 139\n    }\n  ],\n  \"weights\": [\n    {\n      \"course_id\": \"A\",\n      \"prep_plating_load\": 118\n    },\n    {\n      \"course_id\": \"B\",\n      \"prep_plating_load\": 123\n    },\n    {\n      \"course_id\": \"C\",\n      \"prep_plating_load\": 142\n    },\n    {\n      \"course_id\": \"D\",\n      \"prep_plating_load\": 147\n    },\n    {\n      \"course_id\": \"E\",\n      \"prep_plating_load\": 101\n    },\n    {\n      \"course_id\": \"F\",\n      \"prep_plating_load\": 116\n    },\n    {\n      \"course_id\": \"G\",\n      \"prep_plating_load\": 115\n    },\n    {\n      \"course_id\": \"H\",\n      \"prep_plating_load\": 119\n    },\n    {\n      \"course_id\": \"I\",\n      \"prep_plating_load\": 119\n    },\n    {\n      \"course_id\": \"J\",\n      \"prep_plating_load\": 112\n    },\n    {\n      \"course_id\": \"K\",\n      \"prep_plating_load\": 134\n    },\n    {\n      \"course_id\": \"L\",\n      \"prep_plating_load\": 136\n    },\n    {\n      \"course_id\": \"M\",\n      \"prep_plating_load\": 132\n    },\n    {\n      \"course_id\": \"N\",\n      \"prep_plating_load\": 126\n    },\n    {\n      \"course_id\": \"O\",\n      \"prep_plating_load\": 118\n    },\n    {\n      \"course_id\": \"P\",\n      \"prep_plating_load\": 108\n    },\n    {\n      \"course_id\": \"Q\",\n      \"prep_plating_load\": 118\n    },\n    {\n      \"course_id\": \"R\",\n      \"prep_plating_load\": 111\n    },\n    {\n      \"course_id\": \"S\",\n      \"prep_plating_load\": 145\n    },\n    {\n      \"course_id\": \"T\",\n      \"prep_plating_load\": 105\n    },\n    {\n      \"course_id\": \"U\",\n      \"prep_plating_load\": 138\n    },\n    {\n      \"course_id\": \"V\",\n      \"prep_plating_load\": 133\n    }\n  ],\n  \"quadratic\": [\n    {\n      \"course_i_id\": \"A\",\n      \"course_j_id\": \"B\",\n      \"pair_complement_boost\": 125\n    },\n    {\n      \"course_i_id\": \"A\",\n      \"course_j_id\": \"C\",\n      \"pair_complement_boost\": 115\n    },\n    {\n      \"course_i_id\": \"A\",\n      \"course_j_id\": \"E\",\n      \"pair_complement_boost\": 123\n    },\n    {\n      \"course_i_id\": \"A\",\n      \"course_j_id\": \"G\",\n      \"pair_complement_boost\": 125\n    },\n    {\n      \"course_i_id\": \"A\",\n      \"course_j_id\": \"H\",\n      \"pair_complement_boost\": 109\n    },\n    {\n      \"course_i_id\": \"A\",\n      \"course_j_id\": \"I\",\n      \"pair_complement_boost\": 114\n    },\n    {\n      \"course_i_id\": \"A\",\n      \"course_j_id\": \"J\",\n      \"pair_complement_boost\": 129\n    },\n    {\n      \"course_i_id\": \"A\",\n      \"course_j_id\": \"K\",\n      \"pair_complement_boost\": 123\n    },\n    {\n      \"course_i_id\": \"A\",\n      \"course_j_id\": \"L\",\n      \"pair_complement_boost\": 118\n    },\n    {\n      \"course_i_id\": \"A\",\n      \"course_j_id\": \"M\",\n      \"pair_complement_boost\": 121\n    },\n    {\n      \"course_i_id\": \"A\",\n      \"course_j_id\": \"O\",\n      \"pair_complement_boost\": 129\n    },\n    {\n      \"course_i_id\": \"A\",\n      \"course_j_id\": \"P\",\n      \"pair_complement_boost\": 119\n    },\n    {\n      \"course_i_id\": \"A\",\n      \"course_j_id\": \"Q\",\n      \"pair_complement_boost\": 121\n    },\n    {\n      \"course_i_id\": \"A\",\n      \"course_j_id\": \"R\",\n      \"pair_complement_boost\": 110\n    },\n    {\n      \"course_i_id\": \"A\",\n      \"course_j_id\": \"S\",\n      \"pair_complement_boost\": 120\n    },\n    {\n      \"course_i_id\": \"A\",\n      \"course_j_id\": \"T\",\n      \"pair_complement_boost\": 121\n    },\n    {\n      \"course_i_id\": \"A\",\n      \"course_j_id\": \"U\",\n      \"pair_complement_boost\": 127\n    },\n    {\n      \"course_i_id\": \"A\",\n      \"course_j_id\": \"V\",\n      \"pair_complement_boost\": 115\n    },\n    {\n      \"course_i_id\": \"B\",\n      \"course_j_id\": \"C\",\n      \"pair_complement_boost\": 131\n    },\n    {\n      \"course_i_id\": \"B\",\n      \"course_j_id\": \"D\",\n      \"pair_complement_boost\": 111\n    },\n    {\n      \"course_i_id\": \"B\",\n      \"course_j_id\": \"E\",\n      \"pair_complement_boost\": 121\n    },\n    {\n      \"course_i_id\": \"B\",\n      \"course_j_id\": \"F\",\n      \"pair_complement_boost\": 132\n    },\n    {\n      \"course_i_id\": \"B\",\n      \"course_j_id\": \"H\",\n      \"pair_complement_boost\": 118\n    },\n    {\n      \"course_i_id\": \"B\",\n      \"course_j_id\": \"I\",\n      \"pair_complement_boost\": 117\n    },\n    {\n      \"course_i_id\": \"B\",\n      \"course_j_id\": \"J\",\n      \"pair_complement_boost\": 131\n    },\n    {\n      \"course_i_id\": \"B\",\n      \"course_j_id\": \"K\",\n      \"pair_complement_boost\": 123\n    },\n    {\n      \"course_i_id\": \"B\",\n      \"course_j_id\": \"L\",\n      \"pair_complement_boost\": 121\n    },\n    {\n      \"course_i_id\": \"B\",\n      \"course_j_id\": \"N\",\n      \"pair_complement_boost\": 129\n    },\n    {\n      \"course_i_id\": \"B\",\n      \"course_j_id\": \"O\",\n      \"pair_complement_boost\": 130\n    },\n    {\n      \"course_i_id\": \"B\",\n      \"course_j_id\": \"P\",\n      \"pair_complement_boost\": 123\n    },\n    {\n      \"course_i_id\": \"B\",\n      \"course_j_id\": \"Q\",\n      \"pair_complement_boost\": 129\n    },\n    {\n      \"course_i_id\": \"B\",\n      \"course_j_id\": \"S\",\n      \"pair_complement_boost\": 115\n    },\n    {\n      \"course_i_id\": \"B\",\n      \"course_j_id\": \"T\",\n      \"pair_complement_boost\": 117\n    },\n    {\n      \"course_i_id\": \"B\",\n      \"course_j_id\": \"U\",\n      \"pair_complement_boost\": 110\n    },\n    {\n      \"course_i_id\": \"B\",\n      \"course_j_id\": \"V\",\n      \"pair_complement_boost\": 134\n    },\n    {\n      \"course_i_id\": \"C\",\n      \"course_j_id\": \"D\",\n      \"pair_complement_boost\": 128\n    },\n    {\n      \"course_i_id\": \"C\",\n      \"course_j_id\": \"E\",\n      \"pair_complement_boost\": 119\n    },\n    {\n      \"course_i_id\": \"C\",\n      \"course_j_id\": \"G\",\n      \"pair_complement_boost\": 122\n    },\n    {\n      \"course_i_id\": \"C\",\n      \"course_j_id\": \"I\",\n      \"pair_complement_boost\": 128\n    },\n    {\n      \"course_i_id\": \"C\",\n      \"course_j_id\": \"J\",\n      \"pair_complement_boost\": 143\n    },\n    {\n      \"course_i_id\": \"C\",\n      \"course_j_id\": \"K\",\n      \"pair_complement_boost\": 138\n    },\n    {\n      \"course_i_id\": \"C\",\n      \"course_j_id\": \"L\",\n      \"pair_complement_boost\": 129\n    },\n    {\n      \"course_i_id\": \"C\",\n      \"course_j_id\": \"M\",\n      \"pair_complement_boost\": 131\n    },\n    {\n      \"course_i_id\": \"C\",\n      \"course_j_id\": \"N\",\n      \"pair_complement_boost\": 142\n    },\n    {\n      \"course_i_id\": \"C\",\n      \"course_j_id\": \"P\",\n      \"pair_complement_boost\": 120\n    },\n    {\n      \"course_i_id\": \"C\",\n      \"course_j_id\": \"Q\",\n      \"pair_complement_boost\": 132\n    },\n    {\n      \"course_i_id\": \"C\",\n      \"course_j_id\": \"R\",\n      \"pair_complement_boost\": 120\n    },\n    {\n      \"course_i_id\": \"C\",\n      \"course_j_id\": \"S\",\n      \"pair_complement_boost\": 139\n    },\n    {\n      \"course_i_id\": \"C\",\n      \"course_j_id\": \"U\",\n      \"pair_complement_boost\": 128\n    },\n    {\n      \"course_i_id\": \"C\",\n      \"course_j_id\": \"V\",\n      \"pair_complement_boost\": 140\n    },\n    {\n      \"course_i_id\": \"D\",\n      \"course_j_id\": \"E\",\n      \"pair_complement_boost\": 138\n    },\n    {\n      \"course_i_id\": \"D\",\n      \"course_j_id\": \"F\",\n      \"pair_complement_boost\": 144\n    },\n    {\n      \"course_i_id\": \"D\",\n      \"course_j_id\": \"G\",\n      \"pair_complement_boost\": 135\n    },\n    {\n      \"course_i_id\": \"D\",\n      \"course_j_id\": \"H\",\n      \"pair_complement_boost\": 132\n    },\n    {\n      \"course_i_id\": \"D\",\n      \"course_j_id\": \"I\",\n      \"pair_complement_boost\": 141\n    },\n    {\n      \"course_i_id\": \"D\",\n      \"course_j_id\": \"J\",\n      \"pair_complement_boost\": 132\n    },\n    {\n      \"course_i_id\": \"D\",\n      \"course_j_id\": \"K\",\n      \"pair_complement_boost\": 139\n    },\n    {\n      \"course_i_id\": \"D\",\n      \"course_j_id\": \"L\",\n      \"pair_complement_boost\": 133\n    },\n    {\n      \"course_i_id\": \"D\",\n      \"course_j_id\": \"M\",\n      \"pair_complement_boost\": 126\n    },\n    {\n      \"course_i_id\": \"D\",\n      \"course_j_id\": \"N\",\n      \"pair_complement_boost\": 123\n    },\n    {\n      \"course_i_id\": \"D\",\n      \"course_j_id\": \"P\",\n      \"pair_complement_boost\": 140\n    },\n    {\n      \"course_i_id\": \"D\",\n      \"course_j_id\": \"Q\",\n      \"pair_complement_boost\": 147\n    },\n    {\n      \"course_i_id\": \"D\",\n      \"course_j_id\": \"R\",\n      \"pair_complement_boost\": 136\n    },\n    {\n      \"course_i_id\": \"D\",\n      \"course_j_id\": \"S\",\n      \"pair_complement_boost\": 132\n    },\n    {\n      \"course_i_id\": \"D\",\n      \"course_j_id\": \"T\",\n      \"pair_complement_boost\": 141\n    },\n    {\n      \"course_i_id\": \"D\",\n      \"course_j_id\": \"U\",\n      \"pair_complement_boost\": 147\n    },\n    {\n      \"course_i_id\": \"E\",\n      \"course_j_id\": \"G\",\n      \"pair_complement_boost\": 116\n    },\n    {\n      \"course_i_id\": \"E\",\n      \"course_j_id\": \"H\",\n      \"pair_complement_boost\": 109\n    },\n    {\n      \"course_i_id\": \"E\",\n      \"course_j_id\": \"I\",\n      \"pair_complement_boost\": 119\n    },\n    {\n      \"course_i_id\": \"E\",\n      \"course_j_id\": \"J\",\n      \"pair_complement_boost\": 111\n    },\n    {\n      \"course_i_id\": \"E\",\n      \"course_j_id\": \"L\",\n      \"pair_complement_boost\": 117\n    },\n    {\n      \"course_i_id\": \"E\",\n      \"course_j_id\": \"N\",\n      \"pair_complement_boost\": 111\n    },\n    {\n      \"course_i_id\": \"E\",\n      \"course_j_id\": \"O\",\n      \"pair_complement_boost\": 115\n    },\n    {\n      \"course_i_id\": \"E\",\n      \"course_j_id\": \"P\",\n      \"pair_complement_boost\": 122\n    },\n    {\n      \"course_i_id\": \"E\",\n      \"course_j_id\": \"Q\",\n      \"pair_complement_boost\": 120\n    },\n    {\n      \"course_i_id\": \"E\",\n      \"course_j_id\": \"U\",\n      \"pair_complement_boost\": 122\n    },\n    {\n      \"course_i_id\": \"E\",\n      \"course_j_id\": \"V\",\n      \"pair_complement_boost\": 131\n    },\n    {\n      \"course_i_id\": \"F\",\n      \"course_j_id\": \"G\",\n      \"pair_complement_boost\": 127\n    },\n    {\n      \"course_i_id\": \"F\",\n      \"course_j_id\": \"H\",\n      \"pair_complement_boost\": 117\n    },\n    {\n      \"course_i_id\": \"F\",\n      \"course_j_id\": \"I\",\n      \"pair_complement_boost\": 124\n    },\n    {\n      \"course_i_id\": \"F\",\n      \"course_j_id\": \"K\",\n      \"pair_complement_boost\": 118\n    },\n    {\n      \"course_i_id\": \"F\",\n      \"course_j_id\": \"N\",\n      \"pair_complement_boost\": 108\n    },\n    {\n      \"course_i_id\": \"F\",\n      \"course_j_id\": \"P\",\n      \"pair_complement_boost\": 113\n    },\n    {\n      \"course_i_id\": \"F\",\n      \"course_j_id\": \"Q\",\n      \"pair_complement_boost\": 112\n    },\n    {\n      \"course_i_id\": \"F\",\n      \"course_j_id\": \"R\",\n      \"pair_complement_boost\": 110\n    },\n    {\n      \"course_i_id\": \"F\",\n      \"course_j_id\": \"T\",\n      \"pair_complement_boost\": 134\n    },\n    {\n      \"course_i_id\": \"F\",\n      \"course_j_id\": \"V\",\n      \"pair_complement_boost\": 127\n    },\n    {\n      \"course_i_id\": \"G\",\n      \"course_j_id\": \"I\",\n      \"pair_complement_boost\": 112\n    },\n    {\n      \"course_i_id\": \"G\",\n      \"course_j_id\": \"J\",\n      \"pair_complement_boost\": 125\n    },\n    {\n      \"course_i_id\": \"G\",\n      \"course_j_id\": \"K\",\n      \"pair_complement_boost\": 130\n    },\n    {\n      \"course_i_id\": \"G\",\n      \"course_j_id\": \"L\",\n      \"pair_complement_boost\": 125\n    },\n    {\n      \"course_i_id\": \"G\",\n      \"course_j_id\": \"M\",\n      \"pair_complement_boost\": 126\n    },\n    {\n      \"course_i_id\": \"G\",\n      \"course_j_id\": \"O\",\n      \"pair_complement_boost\": 130\n    },\n    {\n      \"course_i_id\": \"G\",\n      \"course_j_id\": \"Q\",\n      \"pair_complement_boost\": 121\n    },\n    {\n      \"course_i_id\": \"G\",\n      \"course_j_id\": \"R\",\n      \"pair_complement_boost\": 130\n    },\n    {\n      \"course_i_id\": \"G\",\n      \"course_j_id\": \"S\",\n      \"pair_complement_boost\": 122\n    },\n    {\n      \"course_i_id\": \"G\",\n      \"course_j_id\": \"T\",\n      \"pair_complement_boost\": 126\n    },\n    {\n      \"course_i_id\": \"G\",\n      \"course_j_id\": \"U\",\n      \"pair_complement_boost\": 126\n    },\n    {\n      \"course_i_id\": \"G\",\n      \"course_j_id\": \"V\",\n      \"pair_complement_boost\": 139\n    },\n    {\n      \"course_i_id\": \"H\",\n      \"course_j_id\": \"J\",\n      \"pair_complement_boost\": 130\n    },\n    {\n      \"course_i_id\": \"H\",\n      \"course_j_id\": \"L\",\n      \"pair_complement_boost\": 132\n    },\n    {\n      \"course_i_id\": \"H\",\n      \"course_j_id\": \"O\",\n      \"pair_complement_boost\": 119\n    },\n    {\n      \"course_i_id\": \"H\",\n      \"course_j_id\": \"P\",\n      \"pair_complement_boost\": 123\n    },\n    {\n      \"course_i_id\": \"H\",\n      \"course_j_id\": \"Q\",\n      \"pair_complement_boost\": 121\n    },\n    {\n      \"course_i_id\": \"H\",\n      \"course_j_id\": \"R\",\n      \"pair_complement_boost\": 107\n    },\n    {\n      \"course_i_id\": \"H\",\n      \"course_j_id\": \"S\",\n      \"pair_complement_boost\": 114\n    },\n    {\n      \"course_i_id\": \"H\",\n      \"course_j_id\": \"U\",\n      \"pair_complement_boost\": 124\n    },\n    {\n      \"course_i_id\": \"I\",\n      \"course_j_id\": \"K\",\n      \"pair_complement_boost\": 124\n    },\n    {\n      \"course_i_id\": \"I\",\n      \"course_j_id\": \"L\",\n      \"pair_complement_boost\": 122\n    },\n    {\n      \"course_i_id\": \"I\",\n      \"course_j_id\": \"M\",\n      \"pair_complement_boost\": 132\n    },\n    {\n      \"course_i_id\": \"I\",\n      \"course_j_id\": \"N\",\n      \"pair_complement_boost\": 130\n    },\n    {\n      \"course_i_id\": \"I\",\n      \"course_j_id\": \"O\",\n      \"pair_complement_boost\": 115\n    },\n    {\n      \"course_i_id\": \"I\",\n      \"course_j_id\": \"P\",\n      \"pair_complement_boost\": 129\n    },\n    {\n      \"course_i_id\": \"I\",\n      \"course_j_id\": \"S\",\n      \"pair_complement_boost\": 108\n    },\n    {\n      \"course_i_id\": \"I\",\n      \"course_j_id\": \"V\",\n      \"pair_complement_boost\": 126\n    },\n    {\n      \"course_i_id\": \"J\",\n      \"course_j_id\": \"K\",\n      \"pair_complement_boost\": 122\n    },\n    {\n      \"course_i_id\": \"J\",\n      \"course_j_id\": \"L\",\n      \"pair_complement_boost\": 128\n    },\n    {\n      \"course_i_id\": \"J\",\n      \"course_j_id\": \"M\",\n      \"pair_complement_boost\": 126\n    },\n    {\n      \"course_i_id\": \"J\",\n      \"course_j_id\": \"N\",\n      \"pair_complement_boost\": 110\n    },\n    {\n      \"course_i_id\": \"J\",\n      \"course_j_id\": \"O\",\n      \"pair_complement_boost\": 108\n    },\n    {\n      \"course_i_id\": \"J\",\n      \"course_j_id\": \"Q\",\n      \"pair_complement_boost\": 128\n    },\n    {\n      \"course_i_id\": \"J\",\n      \"course_j_id\": \"U\",\n      \"pair_complement_boost\": 110\n    },\n    {\n      \"course_i_id\": \"J\",\n      \"course_j_id\": \"V\",\n      \"pair_complement_boost\": 127\n    },\n    {\n      \"course_i_id\": \"K\",\n      \"course_j_id\": \"L\",\n      \"pair_complement_boost\": 133\n    },\n    {\n      \"course_i_id\": \"K\",\n      \"course_j_id\": \"M\",\n      \"pair_complement_boost\": 132\n    },\n    {\n      \"course_i_id\": \"K\",\n      \"course_j_id\": \"O\",\n      \"pair_complement_boost\": 133\n    },\n    {\n      \"course_i_id\": \"K\",\n      \"course_j_id\": \"P\",\n      \"pair_complement_boost\": 132\n    },\n    {\n      \"course_i_id\": \"K\",\n      \"course_j_id\": \"R\",\n      \"pair_complement_boost\": 132\n    },\n    {\n      \"course_i_id\": \"K\",\n      \"course_j_id\": \"T\",\n      \"pair_complement_boost\": 127\n    },\n    {\n      \"course_i_id\": \"K\",\n      \"course_j_id\": \"U\",\n      \"pair_complement_boost\": 120\n    },\n    {\n      \"course_i_id\": \"L\",\n      \"course_j_id\": \"M\",\n      \"pair_complement_boost\": 140\n    },\n    {\n      \"course_i_id\": \"L\",\n   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\"pair_complement_boost\": 131\n    },\n    {\n      \"course_i_id\": \"N\",\n      \"course_j_id\": \"R\",\n      \"pair_complement_boost\": 129\n    },\n    {\n      \"course_i_id\": \"N\",\n      \"course_j_id\": \"S\",\n      \"pair_complement_boost\": 114\n    },\n    {\n      \"course_i_id\": \"N\",\n      \"course_j_id\": \"U\",\n      \"pair_complement_boost\": 131\n    },\n    {\n      \"course_i_id\": \"N\",\n      \"course_j_id\": \"V\",\n      \"pair_complement_boost\": 125\n    },\n    {\n      \"course_i_id\": \"O\",\n      \"course_j_id\": \"P\",\n      \"pair_complement_boost\": 111\n    },\n    {\n      \"course_i_id\": \"O\",\n      \"course_j_id\": \"Q\",\n      \"pair_complement_boost\": 124\n    },\n    {\n      \"course_i_id\": \"O\",\n      \"course_j_id\": \"R\",\n      \"pair_complement_boost\": 102\n    },\n    {\n      \"course_i_id\": \"O\",\n      \"course_j_id\": \"T\",\n      \"pair_complement_boost\": 129\n    },\n    {\n      \"course_i_id\": \"O\",\n      \"course_j_id\": \"U\",\n      \"pair_complement_boost\": 127\n    },\n    {\n      \"course_i_id\": \"O\",\n      \"course_j_id\": \"V\",\n      \"pair_complement_boost\": 119\n    },\n    {\n      \"course_i_id\": \"P\",\n      \"course_j_id\": \"Q\",\n      \"pair_complement_boost\": 119\n    },\n    {\n      \"course_i_id\": \"P\",\n      \"course_j_id\": \"R\",\n      \"pair_complement_boost\": 123\n    },\n    {\n      \"course_i_id\": \"P\",\n      \"course_j_id\": \"S\",\n      \"pair_complement_boost\": 111\n    },\n    {\n      \"course_i_id\": \"P\",\n      \"course_j_id\": \"T\",\n      \"pair_complement_boost\": 114\n    },\n    {\n      \"course_i_id\": \"P\",\n      \"course_j_id\": \"U\",\n      \"pair_complement_boost\": 112\n    },\n    {\n      \"course_i_id\": \"P\",\n      \"course_j_id\": \"V\",\n      \"pair_complement_boost\": 114\n    },\n    {\n      \"course_i_id\": \"Q\",\n      \"course_j_id\": \"T\",\n      \"pair_complement_boost\": 119\n    },\n    {\n      \"course_i_id\": \"Q\",\n      \"course_j_id\": \"U\",\n      \"pair_complement_boost\": 111\n    },\n    {\n      \"course_i_id\": \"Q\",\n      \"course_j_id\": \"V\",\n      \"pair_complement_boost\": 113\n    },\n    {\n      \"course_i_id\": \"R\",\n      \"course_j_id\": \"S\",\n      \"pair_complement_boost\": 128\n    },\n    {\n      \"course_i_id\": \"R\",\n      \"course_j_id\": \"U\",\n      \"pair_complement_boost\": 138\n    },\n    {\n      \"course_i_id\": \"R\",\n      \"course_j_id\": \"V\",\n      \"pair_complement_boost\": 144\n    },\n    {\n      \"course_i_id\": \"S\",\n      \"course_j_id\": \"T\",\n      \"pair_complement_boost\": 130\n    },\n    {\n      \"course_i_id\": \"S\",\n      \"course_j_id\": \"V\",\n      \"pair_complement_boost\": 127\n    },\n    {\n      \"course_i_id\": \"T\",\n      \"course_j_id\": \"U\",\n      \"pair_complement_boost\": 120\n    },\n    {\n      \"course_i_id\": \"U\",\n      \"course_j_id\": \"V\",\n      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Think of it like filling out a tiny form: list the courses you choose in that array, and that’s your answer. This is just a sketch of the shape I expect, not the final selection itself.\n\nPlease make sure you use the exact identifiers from the instance input — don’t rename them or invent new labels.\n- for example: \"Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.\"",
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          "item_j_id": "O",
          "quadratic_profit": 120
        },
        {
          "item_i_id": "N",
          "item_j_id": "Q",
          "quadratic_profit": 131
        },
        {
          "item_i_id": "N",
          "item_j_id": "R",
          "quadratic_profit": 129
        },
        {
          "item_i_id": "N",
          "item_j_id": "S",
          "quadratic_profit": 114
        },
        {
          "item_i_id": "N",
          "item_j_id": "U",
          "quadratic_profit": 131
        },
        {
          "item_i_id": "N",
          "item_j_id": "V",
          "quadratic_profit": 125
        },
        {
          "item_i_id": "O",
          "item_j_id": "O",
          "quadratic_profit": 123
        },
        {
          "item_i_id": "O",
          "item_j_id": "P",
          "quadratic_profit": 111
        },
        {
          "item_i_id": "O",
          "item_j_id": "Q",
          "quadratic_profit": 124
        },
        {
          "item_i_id": "O",
          "item_j_id": "R",
          "quadratic_profit": 102
        },
        {
          "item_i_id": "O",
          "item_j_id": "T",
          "quadratic_profit": 129
        },
        {
          "item_i_id": "O",
          "item_j_id": "U",
          "quadratic_profit": 127
        },
        {
          "item_i_id": "O",
          "item_j_id": "V",
          "quadratic_profit": 119
        },
        {
          "item_i_id": "P",
          "item_j_id": "P",
          "quadratic_profit": 115
        },
        {
          "item_i_id": "P",
          "item_j_id": "Q",
          "quadratic_profit": 119
        },
        {
          "item_i_id": "P",
          "item_j_id": "R",
          "quadratic_profit": 123
        },
        {
          "item_i_id": "P",
          "item_j_id": "S",
          "quadratic_profit": 111
        },
        {
          "item_i_id": "P",
          "item_j_id": "T",
          "quadratic_profit": 114
        },
        {
          "item_i_id": "P",
          "item_j_id": "U",
          "quadratic_profit": 112
        },
        {
          "item_i_id": "P",
          "item_j_id": "V",
          "quadratic_profit": 114
        },
        {
          "item_i_id": "Q",
          "item_j_id": "T",
          "quadratic_profit": 119
        },
        {
          "item_i_id": "Q",
          "item_j_id": "U",
          "quadratic_profit": 111
        },
        {
          "item_i_id": "Q",
          "item_j_id": "V",
          "quadratic_profit": 113
        },
        {
          "item_i_id": "R",
          "item_j_id": "R",
          "quadratic_profit": 113
        },
        {
          "item_i_id": "R",
          "item_j_id": "S",
          "quadratic_profit": 128
        },
        {
          "item_i_id": "R",
          "item_j_id": "U",
          "quadratic_profit": 138
        },
        {
          "item_i_id": "R",
          "item_j_id": "V",
          "quadratic_profit": 144
        },
        {
          "item_i_id": "S",
          "item_j_id": "S",
          "quadratic_profit": 111
        },
        {
          "item_i_id": "S",
          "item_j_id": "T",
          "quadratic_profit": 130
        },
        {
          "item_i_id": "S",
          "item_j_id": "V",
          "quadratic_profit": 127
        },
        {
          "item_i_id": "T",
          "item_j_id": "T",
          "quadratic_profit": 121
        },
        {
          "item_i_id": "T",
          "item_j_id": "U",
          "quadratic_profit": 120
        },
        {
          "item_i_id": "U",
          "item_j_id": "U",
          "quadratic_profit": 120
        },
        {
          "item_i_id": "U",
          "item_j_id": "V",
          "quadratic_profit": 119
        }
      ]
    },
    "solution_variant": [
      "A",
      "B",
      "C",
      "E",
      "J",
      "L",
      "P",
      "Q",
      "V"
    ],
    "context_index": 8,
    "input_format": "json",
    "input_index_base": "names"
  },
  {
    "task_name": "QKP",
    "problem_type": "QKP",
    "instruction": "Recently the design team challenged themselves to build the perfect gift box that feels greater than the sum of its parts but still ships cheaply. Every piece has a weight and a basic value, and some duos create extra value when they’re both included. The task is to pick a subset of items so the total box weight stays under the shipping threshold and the overall perceived value equals adding up each chosen item’s base value plus any bonuses for included pairs; nothing can be split or doubled. The concrete details for the items and the pair bonuses are given below.\n\nThere are 24 available, listed as A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V, W, X, and the shipping threshold is 95160.\n\n| item_ref_for_weight | base_value |\n|---|---|\n| A | 4325 |\n| B | 5357 |\n| C | 6782 |\n| D | 5127 |\n| E | 5539 |\n| F | 4709 |\n| G | 2461 |\n| H | 5824 |\n| I | 4261 |\n| J | 3744 |\n| K | 6734 |\n| L | 1041 |\n| M | 6510 |\n| N | 5249 |\n| O | 6171 |\n| P | 5948 |\n| Q | 6931 |\n| R | 5831 |\n| S | 6722 |\n| T | 3196 |\n| U | 6854 |\n| V | 5299 |\n| W | 2279 |\n| X | 1125 |\n\n| item_ref_for_weight | item_weight |\n|---|---|\n| A | 3675 |\n| B | 4707 |\n| C | 6132 |\n| D | 4477 |\n| E | 4889 |\n| F | 4059 |\n| G | 1811 |\n| H | 5174 |\n| I | 3611 |\n| J | 3094 |\n| K | 6084 |\n| L | 391 |\n| M | 5860 |\n| N | 4599 |\n| O | 5521 |\n| P | 5298 |\n| Q | 6281 |\n| R | 5181 |\n| S | 6072 |\n| T | 2546 |\n| U | 6204 |\n| V | 4649 |\n| W | 1629 |\n| X | 475 |\n\n| pair_item_i | pair_item_j | pair_synergy_value |\n|---|---|---|\n| A | D | 2902 |\n| A | E | 3675 |\n| A | F | 347 |\n| A | G | 3675 |\n| A | I | 1442 |\n| A | J | 3675 |\n| A | K | 1480 |\n| A | O | 1763 |\n| A | Q | 2276 |\n| A | R | 1738 |\n| A | S | 1992 |\n| A | T | 3675 |\n| A | U | 576 |\n| A | V | 2002 |\n| A | W | 2002 |\n| A | X | 946 |\n| B | F | 3156 |\n| B | G | 3677 |\n| B | J | 4707 |\n| B | N | 1763 |\n| B | O | 4707 |\n| B | P | 1629 |\n| B | Q | 946 |\n| B | S | 3008 |\n| B | T | 2458 |\n| B | U | 2116 |\n| B | V | 3261 |\n| B | X | 934 |\n| C | D | 1741 |\n| C | E | 3156 |\n| C | F | 3677 |\n| C | G | 6132 |\n| C | H | 3938 |\n| C | I | 4370 |\n| C | K | 1750 |\n| C | L | 2440 |\n| C | N | 139 |\n| C | O | 1738 |\n| C | P | 2918 |\n| C | Q | 6132 |\n| C | R | 1309 |\n| C | S | 1309 |\n| C | T | 1309 |\n| C | U | 320 |\n| C | V | 1309 |\n| C | W | 1309 |\n| C | X | 1309 |\n| D | E | 936 |\n| D | F | 4477 |\n| D | H | 4477 |\n| D | J | 4477 |\n| D | L | 1999 |\n| D | M | 4477 |\n| D | Q | 4149 |\n| D | R | 4149 |\n| D | S | 4149 |\n| D | T | 1668 |\n| D | V | 282 |\n| D | W | 3322 |\n| E | G | 2980 |\n| E | H | 4889 |\n| E | J | 4889 |\n| E | K | 1629 |\n| E | L | 4889 |\n| E | N | 983 |\n| E | O | 2664 |\n| E | P | 4037 |\n| E | Q | 2085 |\n| E | R | 1963 |\n| E | T | 279 |\n| E | V | 2611 |\n| E | X | 1750 |\n| F | H | 1999 |\n| F | I | 2387 |\n| F | J | 1055 |\n| F | K | 2918 |\n| F | L | 4059 |\n| F | M | 4059 |\n| F | O | 1824 |\n| F | P | 734 |\n| F | Q | 4442 |\n| F | R | 279 |\n| F | S | 4210 |\n| F | T | 159 |\n| F | U | 4442 |\n| F | V | 1480 |\n| F | W | 2440 |\n| F | X | 2333 |\n| G | I | 1811 |\n| G | K | 1577 |\n| G | L | 1811 |\n| G | M | 410 |\n| G | N | 410 |\n| G | O | 410 |\n| G | R | 159 |\n| G | S | 361 |\n| G | T | 410 |\n| G | U | 186 |\n| G | V | 410 |\n| H | I | 1629 |\n| H | J | 2080 |\n| H | L | 1402 |\n| H | M | 2704 |\n| H | N | 2704 |\n| H | O | 1512 |\n| H | Q | 2611 |\n| H | S | 2704 |\n| H | T | 2207 |\n| H | V | 455 |\n| H | W | 2704 |\n| I | J | 576 |\n| I | L | 1824 |\n| I | N | 3156 |\n| I | O | 3971 |\n| I | R | 902 |\n| I | S | 4799 |\n| I | U | 3805 |\n| I | V | 6176 |\n| I | X | 2918 |\n| J | L | 2229 |\n| J | N | 2229 |\n| J | O | 1839 |\n| J | Q | 2207 |\n| J | R | 2229 |\n| J | T | 2229 |\n| J | U | 2229 |\n| J | V | 2229 |\n| J | W | 430 |\n| J | X | 2229 |\n| K | L | 1741 |\n| K | M | 780 |\n| K | N | 1741 |\n| K | P | 1741 |\n| K | Q | 1741 |\n| K | R | 1741 |\n| K | S | 917 |\n| K | U | 1738 |\n| K | W | 3823 |\n| K | X | 1442 |\n| L | M | 1386 |\n| L | O | 2183 |\n| L | P | 2183 |\n| L | Q | 985 |\n| L | R | 2183 |\n| L | T | 2183 |\n| L | U | 2183 |\n| L | X | 3779 |\n| M | N | 347 |\n| M | O | 347 |\n| M | P | 347 |\n| M | S | 347 |\n| M | U | 347 |\n| M | V | 734 |\n| M | X | 734 |\n| N | P | 1859 |\n| N | S | 5876 |\n| N | T | 3444 |\n| N | V | 5876 |\n| N | W | 3645 |\n| N | X | 5290 |\n| O | P | 1480 |\n| O | Q | 985 |\n| O | R | 1480 |\n| O | S | 1480 |\n| O | T | 1480 |\n| O | U | 1480 |\n| O | V | 576 |\n| O | W | 1750 |\n| O | X | 3805 |\n| P | Q | 4339 |\n| P | R | 2207 |\n| P | S | 2207 |\n| P | V | 2207 |\n| P | W | 1055 |\n| P | X | 430 |\n| Q | R | 1750 |\n| Q | S | 2546 |\n| Q | T | 2333 |\n| Q | V | 3872 |\n| Q | W | 2080 |\n| Q | X | 576 |\n| R | U | 455 |\n| R | X | 4458 |\n| S | U | 2333 |\n| S | V | 2333 |\n| S | W | 3911 |\n| S | X | 3529 |\n| T | U | 4135 |\n| T | V | 2440 |\n| T | X | 2080 |\n| U | W | 475 |\n| U | X | 2162 |\n| V | W | 946 |\n| V | X | 2169 |\n\nSelections must maximize the box’s perceived value while keeping total weight under the shipping threshold.\n\nYou can just send back the chosen items in a small JSON snippet like this — it’s just the shape I expect, nothing fancy.\n\n{\n  \"solution\": [<selected_item_id>, <selected_item_id>, ...]\n}\n\nHere \"solution\" is simply a list of the items you want in the gift box. Think of each placeholder as the exact item label from the instance (the things you see in the item list). This block is a sketch of the format I want, not your final answer.\n\nPlease use the item identifiers exactly as they appear in the instance input — no renaming, no made-up labels.  \n- for example: \"Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.\"",
    "instance": {
      "linear_coeffs": [
        4325,
        5357,
        6782,
        5127,
        5539,
        4709,
        2461,
        5824,
        4261,
        3744,
        6734,
        1041,
        6510,
        5249,
        6171,
        5948,
        6931,
        5831,
        6722,
        3196,
        6854,
        5299,
        2279,
        1125
      ],
      "quadratic_coeffs": [
        [
          2137,
          0,
          0,
          2902,
          3675,
          347,
          3675,
          0,
          1442,
          3675,
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          0,
          0,
          0,
          1763,
          0,
          2276,
          1738,
          1992,
          3675,
          576,
          2002,
          2002,
          946
        ],
        [
          0,
          4150,
          0,
          0,
          0,
          3156,
          3677,
          0,
          0,
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          0,
          0,
          0,
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          0,
          3008,
          2458,
          2116,
          3261,
          0,
          934
        ],
        [
          0,
          0,
          5693,
          1741,
          3156,
          3677,
          6132,
          3938,
          4370,
          0,
          1750,
          2440,
          0,
          139,
          1738,
          2918,
          6132,
          1309,
          1309,
          1309,
          320,
          1309,
          1309,
          1309
        ],
        [
          0,
          0,
          0,
          3399,
          936,
          4477,
          0,
          4477,
          0,
          4477,
          0,
          1999,
          4477,
          0,
          0,
          0,
          4149,
          4149,
          4149,
          1668,
          0,
          282,
          3322,
          0
        ],
        [
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          0,
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          0,
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          0,
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        ],
        [
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          0,
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          2387,
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          2333
        ],
        [
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          0,
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        [
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        [
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        ],
        [
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        ],
        [
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        ],
        [
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          0,
          0,
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          0,
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        ],
        [
          0,
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        ],
        [
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        ],
        [
          0,
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        ],
        [
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        ],
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        ],
        [
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        ],
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        [
          0,
          0,
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          0,
          0,
          0,
          0,
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          0,
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        ],
        [
          0,
          0,
          0,
          0,
          0,
          0,
          0,
          0,
          0,
          0,
          0,
          0,
          0,
          0,
          0,
          0,
          0,
          0,
          0,
          0,
          0,
          0,
          475,
          2162
        ],
        [
          0,
          0,
          0,
          0,
          0,
          0,
          0,
          0,
          0,
          0,
          0,
          0,
          0,
          0,
          0,
          0,
          0,
          0,
          0,
          0,
          0,
          139,
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          2169
        ],
        [
          0,
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          0,
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          0,
          0,
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          0,
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        ],
        [
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          0,
          0,
          0,
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          0,
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          0,
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          0,
          0,
          0,
          0,
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          0,
          0
        ]
      ],
      "weights": [
        3675,
        4707,
        6132,
        4477,
        4889,
        4059,
        1811,
        5174,
        3611,
        3094,
        6084,
        391,
        5860,
        4599,
        5521,
        5298,
        6281,
        5181,
        6072,
        2546,
        6204,
        4649,
        1629,
        475
      ],
      "capacity": 95160,
      "solution": [
        0,
        1,
        2,
        3,
        4,
        5,
        7,
        8,
        9,
        10,
        11,
        13,
        14,
        15,
        16,
        17,
        18,
        19,
        20,
        21,
        22,
        23
      ],
      "obj": 507963.0,
      "problem_type": "QKP"
    },
    "solution": [
      0,
      1,
      2,
      3,
      4,
      5,
      7,
      8,
      9,
      10,
      11,
      13,
      14,
      15,
      16,
      17,
      18,
      19,
      20,
      21,
      22,
      23
    ],
    "obj": 507963.0,
    "instance_variant": {
      "problem_type": "QKP",
      "num_items": 24,
      "capacity": 95160,
      "items": [
        "A",
        "B",
        "C",
        "D",
        "E",
        "F",
        "G",
        "H",
        "I",
        "J",
        "K",
        "L",
        "M",
        "N",
        "O",
        "P",
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          "quadratic_profit": 107
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          "quadratic_profit": 105
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          "item_j_id": 22,
          "quadratic_profit": 105
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          "item_j_id": 21,
          "quadratic_profit": 110
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          "quadratic_profit": 101
        },
        {
          "item_i_id": 19,
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      ]
    },
    "solution_variant": [
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      3,
      4,
      5,
      6,
      7,
      9,
      11,
      12,
      14,
      15,
      17,
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    ],
    "context_index": 10,
    "input_format": "csv",
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  },
  {
    "task_name": "QKP",
    "problem_type": "QKP",
    "instruction": "There’s a small run of a collector’s edition to pack, and the team is deciding which deluxe components to slip into the box. Each component brings some standalone appeal, some pairs have extra synergy when included together, and every piece adds to the package’s weight. The job is to pick a selection (no duplicates, and nothing partial) that stays within the shipping capacity and gives the biggest total player value — computed as the sum of each chosen item’s base value plus any extra pair bonuses. The full breakdown of parts, weights and bonuses appears below.\n\nBelow are the 16 available deluxe components 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 and the box's shipping capacity 21047.\n\n| component_id | base_player_value |\n|---|---|\n| 0 | 3459 |\n| 1 | 1874 |\n| 2 | 1229 |\n| 3 | 6850 |\n| 4 | 3939 |\n| 5 | 4038 |\n| 6 | 3472 |\n| 7 | 5941 |\n| 8 | 7038 |\n| 9 | 2523 |\n| 10 | 4402 |\n| 11 | 5443 |\n| 12 | 2944 |\n| 13 | 6925 |\n| 14 | 6396 |\n| 15 | 6400 |\n\n| component_id | component_weight |\n|---|---|\n| 0 | 2809 |\n| 1 | 1224 |\n| 2 | 579 |\n| 3 | 6200 |\n| 4 | 3289 |\n| 5 | 3388 |\n| 6 | 2822 |\n| 7 | 5291 |\n| 8 | 6388 |\n| 9 | 1873 |\n| 10 | 3752 |\n| 11 | 4793 |\n| 12 | 2294 |\n| 13 | 6275 |\n| 14 | 5746 |\n| 15 | 5750 |\n\n| component_a_id | component_b_id | pair_synergy_value |\n|---|---|---|\n| 0 | 1 | 1487 |\n| 0 | 2 | 2443 |\n| 0 | 3 | 3792 |\n| 0 | 4 | 659 |\n| 0 | 5 | 1657 |\n| 0 | 6 | 3862 |\n| 0 | 7 | 4002 |\n| 0 | 8 | 3236 |\n| 0 | 9 | 3159 |\n| 0 | 10 | 1083 |\n| 0 | 11 | 956 |\n| 0 | 12 | 2533 |\n| 0 | 13 | 1339 |\n| 0 | 14 | 2830 |\n| 0 | 15 | 2587 |\n| 1 | 2 | 2729 |\n| 1 | 3 | 2036 |\n| 1 | 4 | 2036 |\n| 1 | 5 | 1384 |\n| 1 | 6 | 1245 |\n| 1 | 7 | 1867 |\n| 1 | 8 | 2104 |\n| 1 | 9 | 1350 |\n| 1 | 10 | 1675 |\n| 1 | 11 | 2589 |\n| 1 | 12 | 479 |\n| 1 | 13 | 413 |\n| 1 | 14 | 2517 |\n| 1 | 15 | 5574 |\n| 2 | 3 | 1292 |\n| 2 | 4 | 1466 |\n| 2 | 5 | 1304 |\n| 2 | 6 | 1272 |\n| 2 | 7 | 608 |\n| 2 | 8 | 1030 |\n| 2 | 9 | 1868 |\n| 2 | 10 | 1493 |\n| 2 | 11 | 1548 |\n| 2 | 12 | 5998 |\n| 2 | 13 | 2009 |\n| 2 | 14 | 3890 |\n| 2 | 15 | 6224 |\n| 3 | 4 | 4779 |\n| 3 | 5 | 4558 |\n| 3 | 6 | 5835 |\n| 3 | 7 | 1896 |\n| 3 | 8 | 3352 |\n| 3 | 9 | 5498 |\n| 3 | 10 | 2945 |\n| 3 | 11 | 5970 |\n| 3 | 12 | 3644 |\n| 3 | 13 | 1722 |\n| 3 | 14 | 2048 |\n| 3 | 15 | 3509 |\n| 4 | 5 | 3482 |\n| 4 | 6 | 1262 |\n| 4 | 7 | 2777 |\n| 4 | 8 | 3643 |\n| 4 | 9 | 4125 |\n| 4 | 10 | 2182 |\n| 4 | 11 | 2626 |\n| 4 | 12 | 2041 |\n| 4 | 13 | 4922 |\n| 4 | 14 | 4635 |\n| 4 | 15 | 4651 |\n| 5 | 6 | 4520 |\n| 5 | 7 | 5643 |\n| 5 | 8 | 4424 |\n| 5 | 9 | 6059 |\n| 5 | 10 | 3370 |\n| 5 | 11 | 5329 |\n| 5 | 12 | 4018 |\n| 5 | 13 | 2462 |\n| 5 | 14 | 6394 |\n| 5 | 15 | 5564 |\n| 6 | 7 | 3956 |\n| 6 | 8 | 2605 |\n| 6 | 9 | 4940 |\n| 6 | 10 | 3991 |\n| 6 | 11 | 1790 |\n| 6 | 12 | 3184 |\n| 6 | 13 | 3675 |\n| 6 | 14 | 4423 |\n| 6 | 15 | 622 |\n| 7 | 8 | 1246 |\n| 7 | 9 | 2181 |\n| 7 | 10 | 1200 |\n| 7 | 11 | 1768 |\n| 7 | 12 | 2754 |\n| 7 | 13 | 1559 |\n| 7 | 14 | 1960 |\n| 7 | 15 | 488 |\n| 8 | 9 | 1319 |\n| 8 | 10 | 1076 |\n| 8 | 11 | 2643 |\n| 8 | 12 | 1049 |\n| 8 | 13 | 3507 |\n| 8 | 14 | 1645 |\n| 8 | 15 | 3113 |\n| 9 | 10 | 909 |\n| 9 | 11 | 2264 |\n| 9 | 12 | 1990 |\n| 9 | 13 | 1064 |\n| 9 | 14 | 1918 |\n| 9 | 15 | 2960 |\n| 10 | 11 | 1762 |\n| 10 | 12 | 2454 |\n| 10 | 13 | 2758 |\n| 10 | 14 | 3323 |\n| 10 | 15 | 3639 |\n| 11 | 12 | 4879 |\n| 11 | 13 | 3383 |\n| 11 | 14 | 3594 |\n| 11 | 15 | 765 |\n| 12 | 13 | 3068 |\n| 12 | 14 | 5708 |\n| 12 | 15 | 5146 |\n| 13 | 14 | 3654 |\n| 13 | 15 | 4464 |\n| 14 | 15 | 4305 |\n\nSelections must respect the 21047 and maximize total player value.\n\nOh, and when you’re ready to send your pick, please use this simple JSON layout so I can read it easily:\n\n{\n  \"solution\": [<component_id>, <component_id>, ...]\n}\n\nThis just means \"solution\" should be a list of the component IDs you want packed into the box. The JSON above is just a sketch of the shape I expect — not the final answer itself.\n\nPlease use the exact identifiers from the instance input — don’t rename them or make up new labels. Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.",
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        6925,
        6396,
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      "quadratic_coeffs": [
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      ],
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      "solution": [
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        6,
        9,
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      ],
      "obj": 112097.0,
      "problem_type": "QKP"
    },
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    ],
    "obj": 112097.0,
    "instance_variant": {
      "problem_type": "QKP",
      "num_items": 16,
      "capacity": 21047,
      "items": [
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        1,
        2,
        3,
        4,
        5,
        6,
        7,
        8,
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      ],
      "linear_pairs": [
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          "linear_profit": 3459
        },
        {
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          "linear_profit": 1874
        },
        {
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          "linear_profit": 1229
        },
        {
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        },
        {
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        },
        {
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        },
        {
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          "linear_profit": 3472
        },
        {
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        },
        {
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        },
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\"photo_i_id\": 9,\n      \"photo_j_id\": 17,\n      \"pairing_bonus\": 100\n    },\n    {\n      \"photo_i_id\": 10,\n      \"photo_j_id\": 13,\n      \"pairing_bonus\": 103\n    },\n    {\n      \"photo_i_id\": 10,\n      \"photo_j_id\": 15,\n      \"pairing_bonus\": 109\n    },\n    {\n      \"photo_i_id\": 11,\n      \"photo_j_id\": 12,\n      \"pairing_bonus\": 106\n    },\n    {\n      \"photo_i_id\": 12,\n      \"photo_j_id\": 13,\n      \"pairing_bonus\": 101\n    },\n    {\n      \"photo_i_id\": 13,\n      \"photo_j_id\": 15,\n      \"pairing_bonus\": 102\n    },\n    {\n      \"photo_i_id\": 16,\n      \"photo_j_id\": 17,\n      \"pairing_bonus\": 109\n    }\n  ]\n}\n\nOh, and when you send back your pick, a little JSON like the one below is perfect — nice and tidy so I can read which photos you chose.\n\n{\n  \"solution\": [<selected_photo_id>, <selected_photo_id>, ...]\n}\n\nNothing fancy: the \"solution\" array is where you list the photo IDs you want in the final spread. This is just a sketch of the shape I expect, not the actual answer — fill it with the exact IDs from the instance when you're ready.\n\nPlease use the identifiers exactly as they appear in the instance input — no renaming and no new labels.\n- for example: \"Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.\"",
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    "task_name": "QKP",
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    "instruction": "Picture this: a camping menu to assemble where each food item improves meal quality a bit on its own, but certain combinations — like fresh bread with good cheese — give an extra morale kick if both make it into the pack. Every item has a weight, you either pack it or you don’t (no halves, no doubles), and the whole load needs to fit under the gear weight limit. The trip’s score is the sum of every chosen item’s base benefit plus any extra bonuses from pairs that travel together, so the job is to pick the set that yields the highest total while staying within the weight allowance. The specific items, their weights, base comforts, and pair bonuses are listed below.\n\nYou'll find 20 food options (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20) and a gear weight limit of 1606 to respect.\n\n| food_id | base_comfort |\n|---|---|\n| 1 | 105 |\n| 2 | 102 |\n| 3 | 105 |\n| 4 | 104 |\n| 5 | 103 |\n| 6 | 103 |\n| 7 | 104 |\n| 8 | 101 |\n| 9 | 105 |\n| 10 | 103 |\n| 11 | 101 |\n| 12 | 103 |\n| 13 | 105 |\n| 14 | 105 |\n| 15 | 102 |\n| 16 | 104 |\n| 17 | 104 |\n| 18 | 102 |\n| 19 | 105 |\n| 20 | 100 |\n\n| food_id | pack_weight |\n|---|---|\n| 1 | 105 |\n| 2 | 100 |\n| 3 | 102 |\n| 4 | 104 |\n| 5 | 101 |\n| 6 | 100 |\n| 7 | 104 |\n| 8 | 102 |\n| 9 | 104 |\n| 10 | 103 |\n| 11 | 101 |\n| 12 | 104 |\n| 13 | 105 |\n| 14 | 105 |\n| 15 | 101 |\n| 16 | 102 |\n| 17 | 102 |\n| 18 | 102 |\n| 19 | 100 |\n| 20 | 101 |\n\n| food_i_id | food_j_id | pair_synergy_bonus |\n|---|---|---|\n| 1 | 4 | 100 |\n| 1 | 5 | 105 |\n| 1 | 6 | 103 |\n| 1 | 8 | 101 |\n| 1 | 11 | 103 |\n| 1 | 12 | 105 |\n| 1 | 13 | 104 |\n| 1 | 14 | 105 |\n| 1 | 16 | 103 |\n| 1 | 18 | 100 |\n| 1 | 19 | 100 |\n| 1 | 20 | 105 |\n| 2 | 3 | 105 |\n| 2 | 6 | 104 |\n| 2 | 7 | 103 |\n| 2 | 9 | 100 |\n| 2 | 10 | 102 |\n| 2 | 11 | 100 |\n| 2 | 12 | 104 |\n| 2 | 13 | 103 |\n| 2 | 16 | 105 |\n| 2 | 17 | 102 |\n| 2 | 18 | 100 |\n| 2 | 19 | 102 |\n| 2 | 20 | 102 |\n| 3 | 4 | 100 |\n| 3 | 6 | 105 |\n| 3 | 11 | 105 |\n| 3 | 12 | 101 |\n| 3 | 14 | 100 |\n| 3 | 15 | 102 |\n| 3 | 18 | 101 |\n| 3 | 20 | 100 |\n| 4 | 5 | 105 |\n| 4 | 6 | 102 |\n| 4 | 7 | 104 |\n| 4 | 8 | 101 |\n| 4 | 12 | 104 |\n| 4 | 14 | 103 |\n| 4 | 18 | 103 |\n| 4 | 19 | 104 |\n| 5 | 7 | 105 |\n| 5 | 10 | 104 |\n| 5 | 11 | 104 |\n| 5 | 12 | 102 |\n| 5 | 19 | 104 |\n| 6 | 7 | 101 |\n| 6 | 8 | 101 |\n| 6 | 10 | 103 |\n| 6 | 13 | 104 |\n| 6 | 15 | 105 |\n| 6 | 17 | 104 |\n| 6 | 18 | 101 |\n| 7 | 9 | 103 |\n| 7 | 12 | 101 |\n| 7 | 14 | 104 |\n| 7 | 17 | 101 |\n| 7 | 19 | 105 |\n| 8 | 11 | 102 |\n| 8 | 12 | 101 |\n| 8 | 14 | 100 |\n| 8 | 18 | 104 |\n| 8 | 19 | 101 |\n| 8 | 20 | 101 |\n| 9 | 12 | 105 |\n| 9 | 14 | 102 |\n| 9 | 15 | 104 |\n| 9 | 18 | 104 |\n| 10 | 12 | 100 |\n| 10 | 13 | 101 |\n| 10 | 17 | 104 |\n| 10 | 19 | 100 |\n| 11 | 12 | 101 |\n| 11 | 13 | 105 |\n| 11 | 14 | 102 |\n| 11 | 15 | 103 |\n| 11 | 18 | 102 |\n| 11 | 19 | 104 |\n| 12 | 14 | 104 |\n| 12 | 16 | 102 |\n| 12 | 17 | 100 |\n| 12 | 18 | 102 |\n| 13 | 14 | 104 |\n| 13 | 15 | 100 |\n| 13 | 16 | 100 |\n| 13 | 18 | 102 |\n| 13 | 19 | 102 |\n| 13 | 20 | 102 |\n| 14 | 16 | 101 |\n| 14 | 17 | 105 |\n| 14 | 20 | 103 |\n| 15 | 16 | 105 |\n| 15 | 19 | 100 |\n| 15 | 20 | 101 |\n| 16 | 19 | 104 |\n| 16 | 20 | 101 |\n| 17 | 18 | 100 |\n| 17 | 20 | 103 |\n| 18 | 20 | 101 |\n\nChoose the set that fits the 1606 limit and maximizes overall meal comfort.\n\nOh, and when you send back the chosen items, please use this simple JSON layout so it's easy to read and check:\n\n{\n  \"solution\": [<selected_item_id>, <selected_item_id>, ...]\n}\n\nThis means: \"solution\" should be an array containing the identifiers of the items you want to pack. Think of it like a short list on a form — each entry is the ID of one selected item. The block above is just the expected shape (a sketch), not the actual answer.\n\nPlease use the item identifiers exactly as they appear in the instance input — no renaming and no new labels.\n- for example: \"Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.\"",
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          "item_j_id": 20,
          "quadratic_profit": 101
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          "quadratic_profit": 103
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          "quadratic_profit": 105
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          "quadratic_profit": 102
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          "quadratic_profit": 104
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          "quadratic_profit": 104
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          "item_j_id": 12,
          "quadratic_profit": 100
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          "quadratic_profit": 101
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          "quadratic_profit": 104
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          "quadratic_profit": 100
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          "quadratic_profit": 101
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          "item_j_id": 12,
          "quadratic_profit": 101
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          "quadratic_profit": 103
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          "quadratic_profit": 102
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          "quadratic_profit": 104
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          "item_j_id": 14,
          "quadratic_profit": 104
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          "quadratic_profit": 102
        },
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          "quadratic_profit": 100
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          "quadratic_profit": 102
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          "quadratic_profit": 101
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          "quadratic_profit": 104
        },
        {
          "item_i_id": 13,
          "item_j_id": 15,
          "quadratic_profit": 100
        },
        {
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          "quadratic_profit": 100
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          "quadratic_profit": 102
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          "quadratic_profit": 102
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          "quadratic_profit": 101
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          "quadratic_profit": 105
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          "quadratic_profit": 103
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          "quadratic_profit": 100
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          "quadratic_profit": 105
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          "quadratic_profit": 100
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          "quadratic_profit": 101
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  {
    "task_name": "QKP",
    "problem_type": "QKP",
    "instruction": "We’re trying to stock a small shelf for the office and want choices that make people happiest. Every supply has a basic usefulness score, and certain combinations create extra perks when both are on the shelf — think tape and a dispenser that together are handier than apart. The plan is to pick individual, non-duplicated items that all fit in the shelf space, then total up their individual usefulness plus any extra pair bonuses to figure out which combination gives the biggest payoff. The concrete item sizes, usefulness numbers, and pair bonuses are listed below.\n\nThere are 24 distinct supplies (listed as 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24), and we have 851 total shelf volume to fill.\nWe get 102 usefulness if we stock 1.\nWe get 102 usefulness if we stock 2.\nWe get 101 usefulness if we stock 3.\nWe get 105 usefulness if we stock 4.\nWe get 100 usefulness if we stock 5.\nWe get 103 usefulness if we stock 6.\nWe get 105 usefulness if we stock 7.\nWe get 104 usefulness if we stock 8.\nWe get 102 usefulness if we stock 9.\nWe get 102 usefulness if we stock 10.\nWe get 102 usefulness if we stock 11.\nWe get 102 usefulness if we stock 12.\nWe get 101 usefulness if we stock 13.\nWe get 101 usefulness if we stock 14.\nWe get 103 usefulness if we stock 15.\nWe get 100 usefulness if we stock 16.\nWe get 101 usefulness if we stock 17.\nWe get 105 usefulness if we stock 18.\nWe get 101 usefulness if we stock 19.\nWe get 104 usefulness if we stock 20.\nWe get 103 usefulness if we stock 21.\nWe get 104 usefulness if we stock 22.\nWe get 105 usefulness if we stock 23.\nWe get 104 usefulness if we stock 24.\n1 occupies 100 shelf volume.\n2 occupies 104 shelf volume.\n3 occupies 103 shelf volume.\n4 occupies 105 shelf volume.\n5 occupies 105 shelf volume.\n6 occupies 102 shelf volume.\n7 occupies 103 shelf volume.\n8 occupies 105 shelf volume.\n9 occupies 105 shelf volume.\n10 occupies 102 shelf volume.\n11 occupies 105 shelf volume.\n12 occupies 103 shelf volume.\n13 occupies 103 shelf volume.\n14 occupies 102 shelf volume.\n15 occupies 102 shelf volume.\n16 occupies 100 shelf volume.\n17 occupies 100 shelf volume.\n18 occupies 104 shelf volume.\n19 occupies 102 shelf volume.\n20 occupies 104 shelf volume.\n21 occupies 103 shelf volume.\n22 occupies 104 shelf volume.\n23 occupies 105 shelf volume.\n24 occupies 101 shelf volume.\nIf we stock both 1 and 2, we gain 103 extra usefulness.\nIf we stock both 1 and 3, we gain 103 extra usefulness.\nIf we stock both 1 and 5, we gain 102 extra usefulness.\nIf we stock both 1 and 6, we gain 104 extra usefulness.\nIf we stock both 1 and 8, we gain 103 extra usefulness.\nIf we stock both 1 and 11, we gain 100 extra usefulness.\nIf we stock both 1 and 13, we gain 105 extra usefulness.\nIf we stock both 1 and 15, we gain 103 extra usefulness.\nIf we stock both 1 and 16, we gain 101 extra usefulness.\nIf we stock both 1 and 20, we gain 105 extra usefulness.\nIf we stock both 1 and 22, we gain 103 extra usefulness.\nIf we stock both 2 and 3, we gain 101 extra usefulness.\nIf we stock both 2 and 9, we gain 102 extra usefulness.\nIf we stock both 2 and 10, we gain 104 extra usefulness.\nIf we stock both 2 and 12, we gain 101 extra usefulness.\nIf we stock both 2 and 15, we gain 105 extra usefulness.\nIf we stock both 2 and 16, we gain 105 extra usefulness.\nIf we stock both 2 and 18, we gain 105 extra usefulness.\nIf we stock both 2 and 21, we gain 100 extra usefulness.\nIf we stock both 2 and 23, we gain 105 extra usefulness.\nIf we stock both 3 and 4, we gain 105 extra usefulness.\nIf we stock both 3 and 5, we gain 105 extra usefulness.\nIf we stock both 3 and 6, we gain 103 extra usefulness.\nIf we stock both 3 and 7, we gain 105 extra usefulness.\nIf we stock both 3 and 10, we gain 102 extra usefulness.\nIf we stock both 3 and 13, we gain 101 extra usefulness.\nIf we stock both 3 and 14, we gain 103 extra usefulness.\nIf we stock both 3 and 15, we gain 102 extra usefulness.\nIf we stock both 3 and 17, we gain 101 extra usefulness.\nIf we stock both 3 and 21, we gain 105 extra usefulness.\nIf we stock both 3 and 22, we gain 100 extra usefulness.\nIf we stock both 3 and 23, we gain 103 extra usefulness.\nIf we stock both 3 and 24, we gain 104 extra usefulness.\nIf we stock both 4 and 5, we gain 101 extra usefulness.\nIf we stock both 4 and 6, we gain 105 extra usefulness.\nIf we stock both 4 and 8, we gain 101 extra usefulness.\nIf we stock both 4 and 9, we gain 102 extra usefulness.\nIf we stock both 4 and 19, we gain 100 extra usefulness.\nIf we stock both 4 and 21, we gain 101 extra usefulness.\nIf we stock both 4 and 23, we gain 100 extra usefulness.\nIf we stock both 5 and 6, we gain 105 extra usefulness.\nIf we stock both 5 and 10, we gain 105 extra usefulness.\nIf we stock both 5 and 11, we gain 102 extra usefulness.\nIf we stock both 5 and 12, we gain 105 extra usefulness.\nIf we stock both 5 and 16, we gain 105 extra usefulness.\nIf we stock both 5 and 18, we gain 100 extra usefulness.\nIf we stock both 5 and 19, we gain 104 extra usefulness.\nIf we stock both 5 and 22, we gain 105 extra usefulness.\nIf we stock both 6 and 10, we gain 101 extra usefulness.\nIf we stock both 6 and 13, we gain 101 extra usefulness.\nIf we stock both 6 and 14, we gain 103 extra usefulness.\nIf we stock both 6 and 17, we gain 102 extra usefulness.\nIf we stock both 6 and 18, we gain 101 extra usefulness.\nIf we stock both 6 and 21, we gain 101 extra usefulness.\nIf we stock both 6 and 22, we gain 104 extra usefulness.\nIf we stock both 7 and 10, we gain 102 extra usefulness.\nIf we stock both 7 and 11, we gain 102 extra usefulness.\nIf we stock both 7 and 13, we gain 105 extra usefulness.\nIf we stock both 7 and 16, we gain 104 extra usefulness.\nIf we stock both 7 and 17, we gain 101 extra usefulness.\nIf we stock both 7 and 20, we gain 100 extra usefulness.\nIf we stock both 7 and 22, we gain 104 extra usefulness.\nIf we stock both 8 and 10, we gain 100 extra usefulness.\nIf we stock both 8 and 11, we gain 103 extra usefulness.\nIf we stock both 8 and 13, we gain 101 extra usefulness.\nIf we stock both 8 and 15, we gain 103 extra usefulness.\nIf we stock both 8 and 16, we gain 104 extra usefulness.\nIf we stock both 8 and 19, we gain 105 extra usefulness.\nIf we stock both 8 and 20, we gain 102 extra usefulness.\nIf we stock both 8 and 24, we gain 103 extra usefulness.\nIf we stock both 9 and 13, we gain 105 extra usefulness.\nIf we stock both 9 and 14, we gain 105 extra usefulness.\nIf we stock both 9 and 18, we gain 100 extra usefulness.\nIf we stock both 9 and 19, we gain 105 extra usefulness.\nIf we stock both 9 and 22, we gain 100 extra usefulness.\nIf we stock both 9 and 23, we gain 105 extra usefulness.\nIf we stock both 10 and 12, we gain 103 extra usefulness.\nIf we stock both 10 and 13, we gain 104 extra usefulness.\nIf we stock both 10 and 14, we gain 105 extra usefulness.\nIf we stock both 10 and 15, we gain 103 extra usefulness.\nIf we stock both 10 and 16, we gain 104 extra usefulness.\nIf we stock both 10 and 21, we gain 103 extra usefulness.\nIf we stock both 10 and 23, we gain 102 extra usefulness.\nIf we stock both 10 and 24, we gain 105 extra usefulness.\nIf we stock both 11 and 12, we gain 103 extra usefulness.\nIf we stock both 11 and 14, we gain 100 extra usefulness.\nIf we stock both 11 and 16, we gain 100 extra usefulness.\nIf we stock both 11 and 18, we gain 102 extra usefulness.\nIf we stock both 11 and 21, we gain 102 extra usefulness.\nIf we stock both 11 and 23, we gain 104 extra usefulness.\nIf we stock both 11 and 24, we gain 101 extra usefulness.\nIf we stock both 12 and 13, we gain 105 extra usefulness.\nIf we stock both 12 and 15, we gain 105 extra usefulness.\nIf we stock both 12 and 16, we gain 100 extra usefulness.\nIf we stock both 12 and 18, we gain 102 extra usefulness.\nIf we stock both 12 and 21, we gain 102 extra usefulness.\nIf we stock both 12 and 22, we gain 101 extra usefulness.\nIf we stock both 12 and 24, we gain 104 extra usefulness.\nIf we stock both 13 and 14, we gain 103 extra usefulness.\nIf we stock both 13 and 15, we gain 104 extra usefulness.\nIf we stock both 13 and 16, we gain 101 extra usefulness.\nIf we stock both 14 and 16, we gain 101 extra usefulness.\nIf we stock both 14 and 19, we gain 100 extra usefulness.\nIf we stock both 14 and 20, we gain 100 extra usefulness.\nIf we stock both 14 and 21, we gain 104 extra usefulness.\nIf we stock both 15 and 16, we gain 104 extra usefulness.\nIf we stock both 15 and 18, we gain 100 extra usefulness.\nIf we stock both 15 and 22, we gain 105 extra usefulness.\nIf we stock both 15 and 23, we gain 101 extra usefulness.\nIf we stock both 16 and 17, we gain 103 extra usefulness.\nIf we stock both 16 and 20, we gain 104 extra usefulness.\nIf we stock both 16 and 21, we gain 104 extra usefulness.\nIf we stock both 16 and 22, we gain 102 extra usefulness.\nIf we stock both 16 and 23, we gain 103 extra usefulness.\nIf we stock both 16 and 24, we gain 104 extra usefulness.\nIf we stock both 17 and 21, we gain 103 extra usefulness.\nIf we stock both 18 and 19, we gain 105 extra usefulness.\nIf we stock both 18 and 21, we gain 105 extra usefulness.\nIf we stock both 18 and 23, we gain 104 extra usefulness.\nIf we stock both 19 and 20, we gain 105 extra usefulness.\nIf we stock both 19 and 21, we gain 100 extra usefulness.\nIf we stock both 19 and 23, we gain 103 extra usefulness.\nIf we stock both 19 and 24, we gain 104 extra usefulness.\nIf we stock both 20 and 24, we gain 102 extra usefulness.\nIf we stock both 22 and 23, we gain 100 extra usefulness.\nIf we stock both 23 and 24, we gain 103 extra usefulness.\nWe'll pick non-duplicated supplies that fit within 851 to maximize overall usefulness.\n\nAlso, when you send back your pick, a tiny JSON snippet like the one below works great — just a single field listing the items you chose.\n\n{\n  \"solution\": [<shelf_item_id>, <shelf_item_id>, ...]\n}\n\nPretty simple: \"solution\" is the list of items you'll put on the shelf, and each <shelf_item_id> placeholder is where you drop an exact item identifier from the instance. Think of it like filling out a short form — the JSON shows the shape I expect, but you should replace those placeholders with the real IDs.\n\nPlease use the identifiers exactly as they appear in the problem input — don't rename them or invent new labels. \nFor example: \"Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.\"",
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    "instruction": "Someone in the after-school program needs to pack activity kits before Monday: every item available contributes a baseline amount of learning and has a certain weight, and a few pairs of items amplify learning when they travel together. The goal is to pick a set of distinct items that together give the highest total learning (add up each chosen item’s value and any extra value from the special pairs) without making the kit heavier than students can handle. Look below for the concrete item weights, individual values, and pair bonuses.\n\n# num_available_items=23\n# max_kit_weight=1783\n# available_item_ids=1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23\n\nitem_identifier,base_learning_value\n1,105\n2,104\n3,101\n4,104\n5,104\n6,105\n7,103\n8,104\n9,101\n10,101\n11,101\n12,103\n13,105\n14,100\n15,104\n16,104\n17,100\n18,100\n19,103\n20,105\n21,102\n22,103\n23,103\n\nitem_identifier,item_weight\n1,101\n2,104\n3,102\n4,105\n5,103\n6,101\n7,100\n8,103\n9,104\n10,105\n11,105\n12,100\n13,104\n14,104\n15,102\n16,100\n17,101\n18,103\n19,103\n20,104\n21,102\n22,105\n23,104\n\nitem_i_identifier,item_j_identifier,interaction_learning_bonus\n1,6,101\n1,9,101\n1,10,100\n1,11,101\n1,12,101\n1,13,101\n1,18,100\n1,19,100\n1,21,101\n1,22,101\n2,3,104\n2,5,102\n2,8,103\n2,10,100\n2,12,100\n2,13,102\n2,15,101\n2,16,100\n2,18,104\n2,20,102\n2,23,101\n3,5,102\n3,8,102\n3,9,100\n3,10,102\n3,12,101\n3,13,102\n3,15,102\n3,17,102\n3,19,102\n3,21,100\n4,8,100\n4,12,105\n4,15,102\n4,16,102\n4,17,102\n4,18,102\n4,19,101\n4,21,102\n4,23,102\n5,8,103\n5,10,103\n5,12,103\n5,13,100\n5,22,103\n5,23,103\n6,9,101\n6,12,104\n6,13,103\n6,14,105\n6,15,100\n6,19,102\n6,21,100\n6,22,102\n7,10,100\n7,13,101\n7,14,101\n7,15,101\n7,17,101\n7,21,101\n7,23,101\n8,10,100\n8,13,100\n8,14,100\n8,15,100\n8,16,100\n8,19,100\n8,23,100\n9,11,100\n9,12,100\n9,13,100\n9,14,101\n9,15,101\n9,18,101\n9,20,101\n10,11,101\n10,15,101\n10,16,101\n10,21,101\n10,23,101\n11,13,100\n11,18,103\n11,19,101\n11,20,103\n11,23,103\n12,13,102\n12,15,101\n12,17,102\n12,19,103\n12,22,104\n13,16,101\n14,15,100\n14,16,105\n14,17,100\n14,20,101\n14,22,102\n14,23,102\n15,17,102\n15,21,101\n15,23,101\n16,17,102\n16,22,100\n16,23,100\n17,23,101\n18,19,102\n18,21,100\n18,22,100\n18,23,100\n19,22,102\n\nAlso, just so we're on the same page, please give your final pick using this simple JSON layout so it's easy to read and check:\n\n{\n  \"solution\": [<selected_item_id>, <selected_item_id>, ...]\n}\n\nThe \"solution\" array should list the item identifiers you want packed in the kit (one identifier per chosen item). Think of it like ticking boxes on a form — put the exact ids of the items you picked inside the brackets. This JSON is just the shape I expect, not your actual answer.\n\nPlease use the item ids exactly as they appear in the instance input — do not rename them or invent new labels. Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.",
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    "task_name": "QKP",
    "problem_type": "QKP",
    "instruction": "On a blank gallery wall, a curator is deciding which prints to hang to create the strongest overall impression within the space available. Each print eats into the wall allowance and offers its own visual value, and select pairs of prints give an added boost when displayed together. The decision is to pick a set of prints that fits inside the allotted area (each print only once) and to tally up the total effect by adding each chosen print’s value plus any bonuses from pairs, aiming for the highest possible total impact. The specific sizes, individual values, and pair bonuses are shown below.\n\nThere are 21 prints available — 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20 — and the curator has 3768 total wall area to allocate.\n\n| print_id | base_visual_value |\n|---|---|\n| 0 | 304 |\n| 1 | 215 |\n| 2 | 318 |\n| 3 | 229 |\n| 4 | 260 |\n| 5 | 184 |\n| 6 | 318 |\n| 7 | 136 |\n| 8 | 143 |\n| 9 | 190 |\n| 10 | 231 |\n| 11 | 223 |\n| 12 | 174 |\n| 13 | 263 |\n| 14 | 322 |\n| 15 | 135 |\n| 16 | 145 |\n| 17 | 180 |\n| 18 | 291 |\n| 19 | 205 |\n| 20 | 179 |\n\n| print_id | display_area |\n|---|---|\n| 0 | 274 |\n| 1 | 185 |\n| 2 | 288 |\n| 3 | 199 |\n| 4 | 230 |\n| 5 | 154 |\n| 6 | 288 |\n| 7 | 106 |\n| 8 | 113 |\n| 9 | 160 |\n| 10 | 201 |\n| 11 | 193 |\n| 12 | 144 |\n| 13 | 233 |\n| 14 | 292 |\n| 15 | 105 |\n| 16 | 115 |\n| 17 | 150 |\n| 18 | 261 |\n| 19 | 175 |\n| 20 | 149 |\n\n| print_i_id | print_j_id | pair_synergy_value |\n|---|---|---|\n| 0 | 2 | 154 |\n| 0 | 6 | 113 |\n| 0 | 8 | 229 |\n| 0 | 10 | 193 |\n| 0 | 12 | 225 |\n| 0 | 13 | 217 |\n| 0 | 14 | 267 |\n| 0 | 15 | 150 |\n| 0 | 17 | 216 |\n| 0 | 18 | 159 |\n| 0 | 20 | 185 |\n| 1 | 2 | 185 |\n| 1 | 3 | 114 |\n| 1 | 5 | 160 |\n| 1 | 6 | 185 |\n| 1 | 7 | 185 |\n| 1 | 8 | 185 |\n| 1 | 9 | 104 |\n| 1 | 11 | 185 |\n| 1 | 12 | 185 |\n| 1 | 14 | 185 |\n| 2 | 6 | 110 |\n| 2 | 7 | 193 |\n| 2 | 11 | 212 |\n| 2 | 12 | 254 |\n| 2 | 13 | 267 |\n| 2 | 17 | 139 |\n| 2 | 19 | 199 |\n| 2 | 20 | 199 |\n| 3 | 4 | 199 |\n| 3 | 5 | 104 |\n| 3 | 7 | 177 |\n| 3 | 9 | 199 |\n| 3 | 11 | 195 |\n| 3 | 12 | 199 |\n| 3 | 14 | 149 |\n| 3 | 16 | 110 |\n| 3 | 17 | 106 |\n| 4 | 6 | 215 |\n| 4 | 7 | 230 |\n| 4 | 11 | 230 |\n| 4 | 12 | 210 |\n| 4 | 16 | 154 |\n| 4 | 19 | 104 |\n| 4 | 20 | 110 |\n| 5 | 8 | 154 |\n| 5 | 11 | 154 |\n| 5 | 16 | 113 |\n| 5 | 18 | 139 |\n| 5 | 19 | 139 |\n| 5 | 20 | 139 |\n| 6 | 9 | 113 |\n| 6 | 12 | 105 |\n| 6 | 14 | 106 |\n| 6 | 15 | 106 |\n| 6 | 16 | 106 |\n| 6 | 17 | 106 |\n| 6 | 20 | 106 |\n| 7 | 8 | 106 |\n| 7 | 12 | 106 |\n| 7 | 14 | 206 |\n| 7 | 15 | 206 |\n| 7 | 16 | 180 |\n| 7 | 17 | 206 |\n| 7 | 19 | 206 |\n| 7 | 20 | 206 |\n| 8 | 10 | 113 |\n| 8 | 12 | 113 |\n| 8 | 13 | 113 |\n| 8 | 15 | 110 |\n| 8 | 18 | 229 |\n| 8 | 19 | 229 |\n| 8 | 20 | 229 |\n| 9 | 12 | 160 |\n| 9 | 17 | 259 |\n| 9 | 18 | 144 |\n| 9 | 20 | 264 |\n| 10 | 12 | 201 |\n| 10 | 14 | 233 |\n| 10 | 16 | 195 |\n| 10 | 18 | 233 |\n| 10 | 19 | 210 |\n| 10 | 20 | 233 |\n| 11 | 13 | 144 |\n| 11 | 15 | 177 |\n| 11 | 16 | 177 |\n| 11 | 17 | 177 |\n| 11 | 18 | 138 |\n| 11 | 20 | 177 |\n| 12 | 14 | 113 |\n| 12 | 20 | 113 |\n| 13 | 15 | 118 |\n| 13 | 16 | 173 |\n| 13 | 17 | 173 |\n| 13 | 18 | 115 |\n| 14 | 17 | 150 |\n| 14 | 20 | 278 |\n| 16 | 18 | 129 |\n| 16 | 19 | 149 |\n| 16 | 20 | 159 |\n\nThe curator will choose a combination of prints that fits the allowance and maximizes the overall visual impact.\n\nAlso, when you tell me which prints you want to hang, just send it back in this simple JSON shape so I can read it easily:\n\n{\n  \"solution\": [<selected_item_id>, <selected_item_id>, ...]\n}\n\n\"solution\" is just the list of prints you picked (use each print's identifier exactly as it appears in the instance). Think of it like filling out a short form: the array holds the ids of the prints you want on the wall. This JSON is only a sketch of the expected shape — not the actual answer.\n\nPlease make sure to use the identifiers exactly as given in the input — no renaming and no new labels.\n\nFor example: \"Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.\"",
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    "problem_type": "QKP",
    "instruction": "Someone on the product team thought it would be fun to treat kit design like a game: which amenities go in to get the biggest cheer from travelers? Every item adds a certain amount of satisfaction and costs a bit of kit weight, and sometimes two items together create a little extra perk that adds more satisfaction on top. The final happiness score is just the sum of the individual item scores plus any pair bonuses, while the practical limits are that the kit’s total weight can’t exceed the packing limit and each amenity is used at most once. The specific item list and pair perks are shown below.\n\nBelow are the 18 amenity options (A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R) and the kit weight capacity is 235.\nAmenity A contributes 106 satisfaction to the kit when included.\nAmenity B contributes 163 satisfaction to the kit when included.\nAmenity C contributes 180 satisfaction to the kit when included.\nAmenity D contributes 124 satisfaction to the kit when included.\nAmenity E contributes 153 satisfaction to the kit when included.\nAmenity F contributes 127 satisfaction to the kit when included.\nAmenity G contributes 119 satisfaction to the kit when included.\nAmenity H contributes 167 satisfaction to the kit when included.\nAmenity I contributes 168 satisfaction to the kit when included.\nAmenity J contributes 186 satisfaction to the kit when included.\nAmenity K contributes 130 satisfaction to the kit when included.\nAmenity L contributes 98 satisfaction to the kit when included.\nAmenity M contributes 145 satisfaction to the kit when included.\nAmenity N contributes 101 satisfaction to the kit when included.\nAmenity O contributes 138 satisfaction to the kit when included.\nAmenity P contributes 109 satisfaction to the kit when included.\nAmenity Q contributes 168 satisfaction to the kit when included.\nAmenity R contributes 140 satisfaction to the kit when included.\nAmenity A has weight 105 toward the kit's packing limit.\nAmenity B has weight 153 toward the kit's packing limit.\nAmenity C has weight 181 toward the kit's packing limit.\nAmenity D has weight 125 toward the kit's packing limit.\nAmenity E has weight 143 toward the kit's packing limit.\nAmenity F has weight 126 toward the kit's packing limit.\nAmenity G has weight 123 toward the kit's packing limit.\nAmenity H has weight 168 toward the kit's packing limit.\nAmenity I has weight 174 toward the kit's packing limit.\nAmenity J has weight 187 toward the kit's packing limit.\nAmenity K has weight 128 toward the kit's packing limit.\nAmenity L has weight 108 toward the kit's packing limit.\nAmenity M has weight 143 toward the kit's packing limit.\nAmenity N has weight 106 toward the kit's packing limit.\nAmenity O has weight 133 toward the kit's packing limit.\nAmenity P has weight 119 toward the kit's packing limit.\nAmenity Q has weight 165 toward the kit's packing limit.\nAmenity R has weight 130 toward the kit's packing limit.\nAmenities A and B together grant an extra 189 satisfaction if both are packed.\nAmenities A and C together grant an extra 179 satisfaction if both are packed.\nAmenities A and D together grant an extra 120 satisfaction if both are packed.\nAmenities A and E together grant an extra 161 satisfaction if both are packed.\nAmenities A and F together grant an extra 180 satisfaction if both are packed.\nAmenities A and G together grant an extra 164 satisfaction if both are packed.\nAmenities A and H together grant an extra 175 satisfaction if both are packed.\nAmenities A and I together grant an extra 136 satisfaction if both are packed.\nAmenities A and J together grant an extra 184 satisfaction if both are packed.\nAmenities A and K together grant an extra 132 satisfaction if both are packed.\nAmenities A and L together grant an extra 134 satisfaction if both are packed.\nAmenities A and M together grant an extra 160 satisfaction if both are packed.\nAmenities A and N together grant an extra 109 satisfaction if both are packed.\nAmenities A and O together grant an extra 138 satisfaction if both are packed.\nAmenities A and P together grant an extra 156 satisfaction if both are packed.\nAmenities A and Q together grant an extra 113 satisfaction if both are packed.\nAmenities A and R together grant an extra 156 satisfaction if both are packed.\nAmenities B and C together grant an extra 157 satisfaction if both are packed.\nAmenities B and D together grant an extra 180 satisfaction if both are packed.\nAmenities B and E together grant an extra 139 satisfaction if both are packed.\nAmenities B and F together grant an extra 152 satisfaction if both are packed.\nAmenities B and G together grant an extra 117 satisfaction if both are packed.\nAmenities B and H together grant an extra 180 satisfaction if both are packed.\nAmenities B and I together grant an extra 161 satisfaction if both are packed.\nAmenities B and J together grant an extra 103 satisfaction if both are packed.\nAmenities B and K together grant an extra 165 satisfaction if both are packed.\nAmenities B and L together grant an extra 182 satisfaction if both are packed.\nAmenities B and M together grant an extra 182 satisfaction if both are packed.\nAmenities B and N together grant an extra 182 satisfaction if both are packed.\nAmenities B and O together grant an extra 175 satisfaction if both are packed.\nAmenities B and P together grant an extra 134 satisfaction if both are packed.\nAmenities B and Q together grant an extra 172 satisfaction if both are packed.\nAmenities B and R together grant an extra 156 satisfaction if both are packed.\nAmenities C and D together grant an extra 104 satisfaction if both are packed.\nAmenities C and E together grant an extra 143 satisfaction if both are packed.\nAmenities C and F together grant an extra 173 satisfaction if both are packed.\nAmenities C and G together grant an extra 178 satisfaction if both are packed.\nAmenities C and H together grant an extra 185 satisfaction if both are packed.\nAmenities C and I together grant an extra 115 satisfaction if both are packed.\nAmenities C and J together grant an extra 145 satisfaction if both are packed.\nAmenities C and K together grant an extra 137 satisfaction if both are packed.\nAmenities C and L together grant an extra 180 satisfaction if both are packed.\nAmenities C and M together grant an extra 196 satisfaction if both are packed.\nAmenities C and N together grant an extra 193 satisfaction if both are packed.\nAmenities C and O together grant an extra 173 satisfaction if both are packed.\nAmenities C and P together grant an extra 132 satisfaction if both are packed.\nAmenities C and Q together grant an extra 177 satisfaction if both are packed.\nAmenities C and R together grant an extra 114 satisfaction if both are packed.\nAmenities D and E together grant an extra 106 satisfaction if both are packed.\nAmenities D and F together grant an extra 141 satisfaction if both are packed.\nAmenities D and G together grant an extra 148 satisfaction if both are packed.\nAmenities D and H together grant an extra 166 satisfaction if both are packed.\nAmenities D and I together grant an extra 124 satisfaction if both are packed.\nAmenities D and J together grant an extra 185 satisfaction if both are packed.\nAmenities D and K together grant an extra 130 satisfaction if both are packed.\nAmenities D and L together grant an extra 153 satisfaction if both are packed.\nAmenities D and M together grant an extra 161 satisfaction if both are packed.\nAmenities D and N together grant an extra 126 satisfaction if both are packed.\nAmenities D and O together grant an extra 186 satisfaction if both are packed.\nAmenities D and P together grant an extra 114 satisfaction if both are packed.\nAmenities D and Q together grant an extra 103 satisfaction if both are packed.\nAmenities D and R together grant an extra 195 satisfaction if both are packed.\nAmenities E and F together grant an extra 113 satisfaction if both are packed.\nAmenities E and G together grant an extra 159 satisfaction if both are packed.\nAmenities E and H together grant an extra 138 satisfaction if both are packed.\nAmenities E and I together grant an extra 136 satisfaction if both are packed.\nAmenities E and J together grant an extra 100 satisfaction if both are packed.\nAmenities E and K together grant an extra 164 satisfaction if both are packed.\nAmenities E and L together grant an extra 104 satisfaction if both are packed.\nAmenities E and M together grant an extra 182 satisfaction if both are packed.\nAmenities E and N together grant an extra 152 satisfaction if both are packed.\nAmenities E and O together grant an extra 147 satisfaction if both are packed.\nAmenities E and P together grant an extra 177 satisfaction if both are packed.\nAmenities E and Q together grant an extra 180 satisfaction if both are packed.\nAmenities E and R together grant an extra 170 satisfaction if both are packed.\nAmenities F and G together grant an extra 100 satisfaction if both are packed.\nAmenities F and H together grant an extra 137 satisfaction if both are packed.\nAmenities F and I together grant an extra 123 satisfaction if both are packed.\nAmenities F and J together grant an extra 133 satisfaction if both are packed.\nAmenities F and K together grant an extra 135 satisfaction if both are packed.\nAmenities F and L together grant an extra 181 satisfaction if both are packed.\nAmenities F and M together grant an extra 144 satisfaction if both are packed.\nAmenities F and N together grant an extra 194 satisfaction if both are packed.\nAmenities F and O together grant an extra 173 satisfaction if both are packed.\nAmenities F and P together grant an extra 115 satisfaction if both are packed.\nAmenities F and Q together grant an extra 187 satisfaction if both are packed.\nAmenities F and R together grant an extra 176 satisfaction if both are packed.\nAmenities G and H together grant an extra 185 satisfaction if both are packed.\nAmenities G and I together grant an extra 196 satisfaction if both are packed.\nAmenities G and J together grant an extra 194 satisfaction if both are packed.\nAmenities G and K together grant an extra 167 satisfaction if both are packed.\nAmenities G and L together grant an extra 165 satisfaction if both are packed.\nAmenities G and M together grant an extra 134 satisfaction if both are packed.\nAmenities G and N together grant an extra 170 satisfaction if both are packed.\nAmenities G and O together grant an extra 117 satisfaction if both are packed.\nAmenities G and P together grant an extra 105 satisfaction if both are packed.\nAmenities G and Q together grant an extra 157 satisfaction if both are packed.\nAmenities G and R together grant an extra 163 satisfaction if both are packed.\nAmenities H and I together grant an extra 110 satisfaction if both are packed.\nAmenities H and J together grant an extra 166 satisfaction if both are packed.\nAmenities H and K together grant an extra 112 satisfaction if both are packed.\nAmenities H and L together grant an extra 107 satisfaction if both are packed.\nAmenities H and M together grant an extra 162 satisfaction if both are packed.\nAmenities H and N together grant an extra 108 satisfaction if both are packed.\nAmenities H and O together grant an extra 197 satisfaction if both are packed.\nAmenities H and P together grant an extra 133 satisfaction if both are packed.\nAmenities H and Q together grant an extra 193 satisfaction if both are packed.\nAmenities H and R together grant an extra 190 satisfaction if both are packed.\nAmenities I and J together grant an extra 129 satisfaction if both are packed.\nAmenities I and K together grant an extra 194 satisfaction if both are packed.\nAmenities I and L together grant an extra 147 satisfaction if both are packed.\nAmenities I and M together grant an extra 123 satisfaction if both are packed.\nAmenities I and N together grant an extra 147 satisfaction if both are packed.\nAmenities I and O together grant an extra 106 satisfaction if both are packed.\nAmenities I and P together grant an extra 132 satisfaction if both are packed.\nAmenities I and Q together grant an extra 165 satisfaction if both are packed.\nAmenities I and R together grant an extra 138 satisfaction if both are packed.\nAmenities J and K together grant an extra 199 satisfaction if both are packed.\nAmenities J and L together grant an extra 173 satisfaction if both are packed.\nAmenities J and M together grant an extra 160 satisfaction if both are packed.\nAmenities J and N together grant an extra 157 satisfaction if both are packed.\nAmenities J and O together grant an extra 182 satisfaction if both are packed.\nAmenities J and P together grant an extra 159 satisfaction if both are packed.\nAmenities J and Q together grant an extra 156 satisfaction if both are packed.\nAmenities J and R together grant an extra 125 satisfaction if both are packed.\nAmenities K and L together grant an extra 185 satisfaction if both are packed.\nAmenities K and M together grant an extra 137 satisfaction if both are packed.\nAmenities K and N together grant an extra 159 satisfaction if both are packed.\nAmenities K and O together grant an extra 167 satisfaction if both are packed.\nAmenities K and P together grant an extra 157 satisfaction if both are packed.\nAmenities K and Q together grant an extra 149 satisfaction if both are packed.\nAmenities K and R together grant an extra 126 satisfaction if both are packed.\nAmenities L and M together grant an extra 151 satisfaction if both are packed.\nAmenities L and N together grant an extra 190 satisfaction if both are packed.\nAmenities L and O together grant an extra 167 satisfaction if both are packed.\nAmenities L and P together grant an extra 127 satisfaction if both are packed.\nAmenities L and Q together grant an extra 191 satisfaction if both are packed.\nAmenities L and R together grant an extra 140 satisfaction if both are packed.\nAmenities M and N together grant an extra 196 satisfaction if both are packed.\nAmenities M and O together grant an extra 128 satisfaction if both are packed.\nAmenities M and P together grant an extra 194 satisfaction if both are packed.\nAmenities M and Q together grant an extra 173 satisfaction if both are packed.\nAmenities M and R together grant an extra 103 satisfaction if both are packed.\nAmenities N and O together grant an extra 138 satisfaction if both are packed.\nAmenities N and P together grant an extra 199 satisfaction if both are packed.\nAmenities N and Q together grant an extra 125 satisfaction if both are packed.\nAmenities N and R together grant an extra 137 satisfaction if both are packed.\nAmenities O and P together grant an extra 152 satisfaction if both are packed.\nAmenities O and Q together grant an extra 134 satisfaction if both are packed.\nAmenities O and R together grant an extra 183 satisfaction if both are packed.\nAmenities P and Q together grant an extra 160 satisfaction if both are packed.\nAmenities P and R together grant an extra 157 satisfaction if both are packed.\nChoose amenities to maximize total satisfaction without exceeding the 235 kit weight.\n\nAlso, when you’re ready to tell me which items you’d pick, just send it back in a tiny JSON snippet like this:\n\n{\n  \"solution\": [<selected_item_id>, <selected_item_id>, ...]\n}\n\nThe \"solution\" array is just the list of amenity IDs you want in the kit — think of it like ticking boxes on a form and listing the exact IDs you 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This JSON is just the shape I expect, not the actual answer itself.\n\nPlease use the identifiers exactly as they appear in the instance input — don’t rename them or invent new labels.\n\nfor example: \"Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.\"",
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        {
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    },
    "solution_variant": [
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    "context_index": 17,
    "input_format": "nl",
    "input_index_base": "names"
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  {
    "task_name": "QKP",
    "problem_type": "QKP",
    "instruction": "We have a limited block of bench time and a menu of possible experiments, each with its own duration and expected contribution to the project. The task is to decide which experiments to actually run so the whole schedule fits the session and the total expected scientific payoff — computed by adding every chosen experiment’s basic value and tacking on any extra value from experiments that synergize with each other — ends up as large as possible. Each experiment can be run at most once and total time can’t exceed the session. The concrete details follow below.\n\n# num_experiments=23\n# session_time_limit=1730\n# experiment_ids=A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V, W\n\nexperiment_id_for_time,base_scientific_value\nA,115\nB,113\nC,115\nD,110\nE,113\nF,111\nG,112\nH,111\nI,114\nJ,110\nK,111\nL,110\nM,115\nN,114\nO,114\nP,112\nQ,110\nR,115\nS,112\nT,112\nU,113\nV,114\nW,111\n\nexperiment_id_for_time,duration\nA,105\nB,103\nC,105\nD,100\nE,103\nF,101\nG,102\nH,101\nI,104\nJ,100\nK,101\nL,100\nM,105\nN,104\nO,104\nP,102\nQ,100\nR,105\nS,102\nT,102\nU,103\nV,104\nW,101\n\nexperiment_i_id,experiment_j_id,pair_synergy_value\nA,B,100\nA,C,104\nA,D,105\nA,E,101\nA,F,104\nA,G,100\nA,H,102\nA,I,103\nA,J,101\nA,K,102\nA,L,105\nA,M,105\nA,N,103\nA,O,102\nA,P,102\nA,Q,102\nA,R,102\nA,S,101\nA,T,101\nA,U,100\nA,V,104\nA,W,102\nB,C,101\nB,D,101\nB,E,103\nB,F,100\nB,G,100\nB,H,103\nB,I,103\nB,J,101\nB,K,100\nB,L,100\nB,M,103\nB,N,103\nB,O,100\nB,P,103\nB,Q,103\nB,R,103\nB,S,103\nB,T,101\nB,U,103\nB,V,102\nB,W,100\nC,D,100\nC,E,100\nC,F,104\nC,G,104\nC,H,103\nC,I,101\nC,J,102\nC,K,103\nC,L,101\nC,M,101\nC,N,103\nC,O,100\nC,P,100\nC,Q,101\nC,R,105\nC,S,101\nC,T,102\nC,U,100\nC,V,102\nC,W,101\nD,E,100\nD,F,100\nD,G,100\nD,H,100\nD,I,100\nD,J,100\nD,K,100\nD,L,100\nD,M,100\nD,N,100\nD,O,100\nD,P,100\nD,Q,100\nD,R,100\nD,S,100\nD,T,100\nD,U,102\nD,V,103\nD,W,100\nE,F,103\nE,G,103\nE,H,103\nE,I,103\nE,J,102\nE,K,103\nE,L,103\nE,M,103\nE,N,103\nE,O,102\nE,P,103\nE,Q,101\nE,R,100\nE,S,103\nE,T,100\nE,U,100\nE,V,100\nE,W,100\nF,G,101\nF,H,101\nF,I,101\nF,J,101\nF,K,101\nF,L,101\nF,M,100\nF,N,101\nF,O,101\nF,P,101\nF,Q,101\nF,R,101\nF,S,100\nF,T,102\nF,U,102\nF,V,100\nF,W,101\nG,H,101\nG,I,100\nG,J,102\nG,K,101\nG,L,102\nG,M,102\nG,N,102\nG,O,102\nG,P,102\nG,Q,102\nG,R,101\nG,S,100\nG,T,100\nG,U,100\nG,V,100\nG,W,100\nH,I,101\nH,J,101\nH,K,101\nH,L,101\nH,M,101\nH,N,101\nH,O,101\nH,P,101\nH,Q,101\nH,R,102\nH,S,101\nH,T,100\nH,U,100\nH,V,102\nH,W,102\nI,J,104\nI,K,100\nI,L,104\nI,M,103\nI,N,100\nI,O,100\nI,P,102\nI,Q,102\nI,R,102\nI,S,100\nI,T,102\nI,U,102\nI,V,102\nI,W,100\nJ,K,100\nJ,L,100\nJ,M,100\nJ,N,100\nJ,O,100\nJ,P,100\nJ,Q,100\nJ,R,100\nJ,S,100\nJ,T,100\nJ,U,100\nJ,V,100\nJ,W,100\nK,L,101\nK,M,101\nK,N,101\nK,O,101\nK,P,101\nK,Q,100\nK,R,100\nK,S,100\nK,T,100\nK,U,100\nK,V,100\nK,W,100\nL,M,100\nL,N,100\nL,O,100\nL,P,100\nL,Q,101\nL,R,100\nL,S,101\nL,T,101\nL,U,101\nL,V,101\nL,W,100\nM,N,104\nM,O,104\nM,P,102\nM,Q,104\nM,R,104\nM,S,101\nM,T,103\nM,U,102\nM,V,104\nM,W,101\nN,O,102\nN,P,103\nN,Q,102\nN,R,103\nN,S,100\nN,T,103\nN,U,100\nN,V,103\nN,W,103\nO,P,102\nO,Q,102\nO,R,102\nO,S,101\nO,T,101\nO,U,100\nO,V,104\nO,W,102\nP,Q,103\nP,R,104\nP,S,100\nP,T,100\nP,U,103\nP,V,101\nP,W,103\nQ,R,102\nQ,S,102\nQ,T,100\nQ,U,102\nQ,V,100\nQ,W,100\nR,S,101\nR,T,102\nR,U,103\nR,V,102\nR,W,103\nS,T,104\nS,U,101\nS,V,102\nS,W,101\nT,U,104\nT,V,101\nT,W,102\nU,V,102\nU,W,103\nV,W,100\n\nAlso, when you send your selection back, please use this simple JSON layout so it's easy to parse:\n\n{\n  \"solution\": [<experiment_id>, <experiment_id>, ...]\n}\n\n\"solution\" is just the list of experiment IDs you plan to run during the session — put each experiment's identifier inside the brackets, separated by commas. This is only a sketch of the shape I expect, not the actual answer.\n\nPlease make sure to use the exact identifiers from the instance input — don't rename them or invent new labels.\n- for example: \"Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.\"",
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    "instruction": "In a small team meeting the problem was simple: there’s one allotted airtime window and many creative options, each taking some seconds and delivering a baseline impact. On top of that, some creatives amplify each other when paired, adding extra impact only if both are scheduled. The task is to pick which creatives to run so their total time stays within the bought slot, no creative is repeated, and the campaign’s total impact — the sum of all chosen creatives’ base impacts plus any applicable pair boosts — comes out as large as possible. The specific asset data and pair boosts are provided below.\n\n# total_assets_count=25\n# total_airtime_seconds=4427\n# asset_identifiers=A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V, W, X, Y\n\nasset_id,baseline_impact\nA,120\nB,154\nC,133\nD,197\nE,190\nF,265\nG,130\nH,173\nI,123\nJ,123\nK,228\nL,277\nM,168\nN,285\nO,199\nP,183\nQ,166\nR,278\nS,201\nT,242\nU,172\nV,243\nW,166\nX,105\nY,179\n\nasset_id,airtime_seconds\nA,112\nB,139\nC,130\nD,203\nE,193\nF,256\nG,140\nH,188\nI,133\nJ,111\nK,223\nL,286\nM,165\nN,273\nO,203\nP,168\nQ,157\nR,293\nS,197\nT,254\nU,158\nV,229\nW,168\nX,109\nY,166\n\nasset_i_id,asset_j_id,pair_bonus_impact\nA,B,112\nA,C,112\nA,H,112\nA,K,112\nA,L,112\nA,M,112\nA,N,112\nA,O,112\nA,P,112\nA,S,112\nA,T,112\nA,W,109\nA,Y,192\nB,C,139\nB,G,123\nB,H,139\nB,P,139\nB,Q,139\nB,R,139\nB,T,139\nB,X,107\nB,Y,154\nC,F,130\nC,G,130\nC,J,130\nC,L,130\nC,M,130\nC,P,130\nC,R,130\nC,S,130\nC,T,130\nC,U,130\nC,V,162\nC,W,193\nC,X,203\nC,Y,203\nD,F,203\nD,J,203\nD,M,201\nD,N,157\nD,P,150\nD,S,198\nD,T,107\nD,U,162\nD,W,140\nE,H,110\nE,I,193\nE,J,193\nE,K,193\nE,L,193\nE,S,134\nE,T,140\nE,U,123\nE,V,256\nE,X,256\nF,G,142\nF,H,209\nF,P,197\nF,Q,254\nF,R,138\nF,S,166\nF,Y,140\nG,K,140\nG,L,140\nG,P,140\nG,R,140\nG,S,108\nG,T,123\nG,V,129\nG,W,247\nH,J,183\nH,K,188\nH,L,188\nH,N,188\nH,P,188\nH,Q,158\nH,R,188\nH,V,123\nI,J,133\nI,L,133\nI,M,133\nI,N,133\nI,O,133\nI,S,138\nI,U,204\nI,Y,158\nJ,L,111\nJ,M,111\nJ,O,111\nJ,P,111\nJ,R,165\nJ,S,241\nJ,U,241\nJ,V,220\nJ,W,108\nK,S,227\nK,U,227\nK,V,180\nK,X,227\nK,Y,138\nL,N,165\nL,R,165\nL,S,147\nM,N,121\nM,Q,142\nM,S,185\nM,T,243\nM,W,107\nM,X,117\nM,Y,183\nN,P,142\nN,Q,183\nN,R,156\nN,S,183\nN,V,107\nO,R,168\nO,S,158\nO,U,158\nO,W,158\nO,X,158\nP,V,147\nP,W,157\nP,X,157\nP,Y,157\nQ,S,159\nQ,T,147\nQ,U,234\nQ,V,138\nQ,X,177\nR,S,156\nR,V,158\nR,Y,117\nS,T,138\nS,Y,158\nT,U,107\nT,X,109\n\nOh, and one more practical thing — when you send back the picks, please use a tiny JSON snippet so it's easy to parse. Something like this:\n\n{\n  \"solution\": [<selected_creative_id>, <selected_creative_id>, ...]\n}\n\nThis just means \"solution\" should be an array listing the IDs of the creatives you chose. Think of it as filling in a short form: put each chosen creative's identifier inside the brackets, separated by commas. The JSON above is just a sketch of the shape I need, not your final answer.\n\nUse the identifiers exactly as they appear in the instance input — don't rename them or invent new labels.\n- for example: \"Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.",
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      "J",
      "K",
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  {
    "task_name": "QKP",
    "problem_type": "QKP",
    "instruction": "I run the catering deliveries for a small company, juggling which platters to load into the van for each event. Each platter has a weight and a base customer appeal, and certain pairs just click together and add extra appeal when both are included. The trick is to pick a set of platters whose combined weight fits in the van and that gives the highest total appeal — total appeal being the sum of each platter’s base appeal plus any extra bonuses from included platter pairs — and nobody gets duplicates of the same platter. The exact platter weights, base appeals, van capacity and pair bonuses are listed below.\n\n# num_platters=21\n# van_capacity=3890\n# platter_ids=0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20\n\nplatter_id,base_appeal\n0,192\n1,234\n2,117\n3,255\n4,240\n5,168\n6,163\n7,128\n8,144\n9,112\n10,154\n11,205\n12,226\n13,275\n14,205\n15,281\n16,208\n17,254\n18,292\n19,187\n20,104\n\nplatter_id,platter_weight\n0,138\n1,248\n2,206\n3,174\n4,291\n5,111\n6,237\n7,165\n8,112\n9,150\n10,243\n11,255\n12,114\n13,233\n14,177\n15,248\n16,183\n17,142\n18,292\n19,182\n20,113\n\nplatter_i_id,platter_j_id,pair_bonus_appeal\n0,1,139\n0,2,149\n0,3,102\n0,4,234\n0,5,123\n0,6,277\n0,7,262\n0,8,286\n0,9,182\n0,10,293\n0,11,102\n0,12,229\n0,13,134\n0,14,141\n0,15,172\n0,16,120\n0,17,197\n0,18,274\n0,19,204\n0,20,234\n1,2,103\n1,3,189\n1,4,100\n1,5,266\n1,6,208\n1,7,189\n1,8,134\n1,9,108\n1,10,142\n1,11,175\n1,12,100\n1,13,198\n1,14,226\n1,15,235\n1,16,271\n1,17,178\n1,18,287\n1,19,205\n1,20,267\n2,3,233\n2,4,169\n2,5,262\n2,6,225\n2,7,142\n2,8,174\n2,9,262\n2,10,151\n2,11,268\n2,12,182\n2,13,160\n2,14,240\n2,15,252\n2,16,182\n2,17,187\n2,18,277\n2,19,259\n2,20,124\n3,4,215\n3,5,198\n3,6,119\n3,7,243\n3,8,146\n3,9,205\n3,10,263\n3,11,109\n3,12,294\n3,13,148\n3,14,152\n3,15,136\n3,16,124\n3,17,245\n3,18,225\n3,19,240\n3,20,182\n4,5,158\n4,6,155\n4,7,233\n4,8,150\n4,9,142\n4,10,252\n4,11,115\n4,12,275\n4,13,241\n4,14,230\n4,15,260\n4,16,165\n4,17,181\n4,18,286\n4,19,253\n4,20,217\n5,6,177\n5,7,289\n5,8,287\n5,9,167\n5,10,120\n5,11,135\n5,12,229\n5,13,278\n5,14,242\n5,15,181\n5,16,239\n5,17,212\n5,18,190\n5,19,173\n5,20,112\n6,7,222\n6,8,171\n6,9,128\n6,10,231\n6,11,110\n6,12,106\n6,13,267\n6,14,248\n6,15,211\n6,16,153\n6,17,200\n6,18,165\n6,19,242\n6,20,196\n7,8,290\n7,9,237\n7,10,165\n7,11,149\n7,12,102\n7,13,197\n7,14,172\n7,15,265\n7,16,224\n7,17,179\n7,18,271\n7,19,105\n7,20,111\n8,9,238\n8,10,152\n8,11,184\n8,12,291\n8,13,205\n8,14,249\n8,15,183\n8,16,242\n8,17,112\n8,18,211\n8,19,187\n8,20,212\n9,10,199\n9,11,250\n9,12,230\n9,13,206\n9,14,132\n9,15,166\n9,16,115\n9,17,211\n9,18,264\n9,19,227\n9,20,132\n10,11,183\n10,12,134\n10,13,222\n10,14,275\n10,15,138\n10,16,108\n10,17,147\n10,18,221\n10,19,114\n10,20,144\n11,12,102\n11,13,205\n11,14,255\n11,15,156\n11,16,103\n11,17,199\n11,18,158\n11,19,125\n11,20,267\n12,13,159\n12,14,203\n12,15,116\n12,16,216\n12,17,237\n12,18,173\n12,19,218\n12,20,256\n13,14,103\n13,15,108\n13,16,187\n13,17,292\n13,18,267\n13,19,153\n13,20,233\n14,15,183\n14,16,108\n14,17,147\n14,18,212\n14,19,208\n14,20,269\n15,16,214\n15,17,164\n15,18,155\n15,19,196\n15,20,268\n16,17,163\n16,18,159\n16,19,152\n16,20,189\n17,18,153\n17,19,255\n17,20,136\n18,19,132\n18,20,169\n\nWhen you tell me which platters to load, just give it in a tiny JSON like this so it's easy to read and check:\n\n{\n  \"solution\": [<selected_platter_id>, <selected_platter_id>, ...]\n}\n\nThis just means \"solution\" is a list of the platters you pick for the van — one identifier per platter. Think of it like ticking boxes on a short form: put each platter's exact ID in the array in whatever order you like. This is only a sketch of the shape I expect, not the actual selection.\n\nPlease use the platter/item identifiers exactly as they appear in the problem input — don't rename them or invent new ones. Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.",
    "instance": {
      "linear_coeffs": [
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        234,
        117,
        255,
        240,
        168,
        163,
        128,
        144,
        112,
        154,
        205,
        226,
        275,
        205,
        281,
        208,
        254,
        292,
        187,
        104
      ],
      "quadratic_coeffs": [
        [
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          234,
          123,
          277,
          262,
          286,
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          293,
          102,
          229,
          134,
          141,
          172,
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          197,
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          204,
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        [
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          103,
          189,
          100,
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          208,
          189,
          134,
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          142,
          175,
          100,
          198,
          226,
          235,
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  {
    "task_name": "QKP",
    "problem_type": "QKP",
    "instruction": "Many people working on the festival have suggested different combos of bands, but each band comes with its own cost and solo drawing power and some particular pairings give a notable extra boost when both are included. The challenge is picking a subset of bands whose combined fees fit the budget, and then measuring success by the total crowd: add each hired band’s individual draw and every extra bonus for pairs that are both in the lineup. The lineup that produces the biggest summed crowd is the one to go with, with the rule that no band is hired more than once and the total fees can’t exceed available funds. The exact cost and boost numbers are listed below.\n\n{\n  \"num_bands\": 20,\n  \"budget_limit\": 1014,\n  \"band_ids\": [\n    1,\n    2,\n    3,\n    4,\n    5,\n    6,\n    7,\n    8,\n    9,\n    10,\n    11,\n    12,\n    13,\n    14,\n    15,\n    16,\n    17,\n    18,\n    19,\n    20\n  ],\n  \"linear\": [\n    {\n      \"band_id\": 1,\n      \"solo_draw\": 130\n    },\n    {\n      \"band_id\": 2,\n      \"solo_draw\": 107\n    },\n    {\n      \"band_id\": 3,\n      \"solo_draw\": 109\n    },\n    {\n      \"band_id\": 4,\n      \"solo_draw\": 147\n    },\n    {\n      \"band_id\": 5,\n      \"solo_draw\": 142\n    },\n    {\n      \"band_id\": 6,\n      \"solo_draw\": 137\n    },\n    {\n      \"band_id\": 7,\n      \"solo_draw\": 137\n    },\n    {\n      \"band_id\": 8,\n      \"solo_draw\": 136\n    },\n    {\n      \"band_id\": 9,\n      \"solo_draw\": 98\n    },\n    {\n      \"band_id\": 10,\n      \"solo_draw\": 143\n    },\n    {\n      \"band_id\": 11,\n      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},\n    {\n      \"band_i_id\": 6,\n      \"band_j_id\": 17,\n      \"pairing_boost\": 122\n    },\n    {\n      \"band_i_id\": 6,\n      \"band_j_id\": 18,\n      \"pairing_boost\": 134\n    },\n    {\n      \"band_i_id\": 6,\n      \"band_j_id\": 19,\n      \"pairing_boost\": 108\n    },\n    {\n      \"band_i_id\": 6,\n      \"band_j_id\": 20,\n      \"pairing_boost\": 108\n    },\n    {\n      \"band_i_id\": 7,\n      \"band_j_id\": 8,\n      \"pairing_boost\": 127\n    },\n    {\n      \"band_i_id\": 7,\n      \"band_j_id\": 10,\n      \"pairing_boost\": 124\n    },\n    {\n      \"band_i_id\": 7,\n      \"band_j_id\": 11,\n      \"pairing_boost\": 132\n    },\n    {\n      \"band_i_id\": 7,\n      \"band_j_id\": 12,\n      \"pairing_boost\": 125\n    },\n    {\n      \"band_i_id\": 7,\n      \"band_j_id\": 13,\n      \"pairing_boost\": 130\n    },\n    {\n      \"band_i_id\": 7,\n      \"band_j_id\": 15,\n      \"pairing_boost\": 118\n    },\n    {\n      \"band_i_id\": 7,\n      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\"pairing_boost\": 134\n    },\n    {\n      \"band_i_id\": 13,\n      \"band_j_id\": 17,\n      \"pairing_boost\": 122\n    },\n    {\n      \"band_i_id\": 13,\n      \"band_j_id\": 18,\n      \"pairing_boost\": 134\n    },\n    {\n      \"band_i_id\": 13,\n      \"band_j_id\": 19,\n      \"pairing_boost\": 143\n    },\n    {\n      \"band_i_id\": 13,\n      \"band_j_id\": 20,\n      \"pairing_boost\": 135\n    },\n    {\n      \"band_i_id\": 14,\n      \"band_j_id\": 15,\n      \"pairing_boost\": 127\n    },\n    {\n      \"band_i_id\": 14,\n      \"band_j_id\": 18,\n      \"pairing_boost\": 136\n    },\n    {\n      \"band_i_id\": 14,\n      \"band_j_id\": 19,\n      \"pairing_boost\": 141\n    },\n    {\n      \"band_i_id\": 14,\n      \"band_j_id\": 20,\n      \"pairing_boost\": 134\n    },\n    {\n      \"band_i_id\": 15,\n      \"band_j_id\": 17,\n      \"pairing_boost\": 129\n    },\n    {\n      \"band_i_id\": 16,\n      \"band_j_id\": 17,\n      \"pairing_boost\": 132\n    },\n    {\n      \"band_i_id\": 16,\n      \"band_j_id\": 18,\n      \"pairing_boost\": 122\n    },\n    {\n      \"band_i_id\": 16,\n      \"band_j_id\": 19,\n      \"pairing_boost\": 136\n    },\n    {\n      \"band_i_id\": 16,\n      \"band_j_id\": 20,\n      \"pairing_boost\": 122\n    },\n    {\n      \"band_i_id\": 17,\n      \"band_j_id\": 19,\n      \"pairing_boost\": 119\n    },\n    {\n      \"band_i_id\": 17,\n      \"band_j_id\": 20,\n      \"pairing_boost\": 136\n    },\n    {\n      \"band_i_id\": 18,\n      \"band_j_id\": 19,\n      \"pairing_boost\": 133\n    }\n  ]\n}\n\nOh, and when you send back the final lineup, please use this simple JSON shape so it's easy to read:\n\n{\n  \"solution\": [<selected_band_id>, <selected_band_id>, ...]\n}\n\nThis \"solution\" array should just list the IDs of the bands you want to hire (one entry per band). Think of it like filling in a form: put the band identifiers in the list, and that's your proposed lineup. The JSON above is only a sketch of the shape I need, not the actual answer.\n\nTo be clear, valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”. Please use the identifiers exactly as they appear in the instance input—no renaming and no new labels.",
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      "problem_type": "QKP"
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    "instance_variant": {
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    "instruction": "On a sunny morning the volunteers sorted through plant tags, figuring out what to put in the community bed. Each tag lists how much room a plant needs and how lovely it looks by itself, and some pairs of plants enhance one another, giving extra beauty if both are chosen. The job is to choose a combination that fits the bed’s available area and yields the greatest total beauty — add every chosen plant’s beauty and then any extra points for pairs — and each plant choice can only be used once. The exact plant items, areas, and pair bonuses are detailed below.\n\nThey examined 25 plant tags — A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V, W, X, Y — for the community bed, which has 1752 area.\n\n| plant_id | base_beauty |\n|---|---|\n| A | 122 |\n| B | 115 |\n| C | 134 |\n| D | 119 |\n| E | 118 |\n| F | 129 |\n| G | 128 |\n| H | 124 |\n| I | 127 |\n| J | 124 |\n| K | 119 |\n| L | 116 |\n| M | 123 |\n| N | 129 |\n| O | 112 |\n| P | 123 |\n| Q | 127 |\n| R | 114 |\n| S | 113 |\n| T | 117 |\n| U | 114 |\n| V | 121 |\n| W | 123 |\n| X | 132 |\n| Y | 117 |\n\n| plant_id | area_required |\n|---|---|\n| A | 110 |\n| B | 103 |\n| C | 122 |\n| D | 107 |\n| E | 106 |\n| F | 117 |\n| G | 116 |\n| H | 112 |\n| I | 115 |\n| J | 112 |\n| K | 107 |\n| L | 104 |\n| M | 111 |\n| N | 117 |\n| O | 100 |\n| P | 111 |\n| Q | 115 |\n| R | 102 |\n| S | 101 |\n| T | 105 |\n| U | 102 |\n| V | 109 |\n| W | 111 |\n| X | 120 |\n| Y | 105 |\n\n| plant_i_id | plant_j_id | pair_bonus_beauty |\n|---|---|---|\n| A | C | 121 |\n| A | D | 124 |\n| A | G | 111 |\n| A | H | 121 |\n| A | K | 123 |\n| A | L | 108 |\n| A | M | 103 |\n| A | Q | 121 |\n| A | R | 100 |\n| A | U | 117 |\n| A | V | 104 |\n| A | X | 101 |\n| B | C | 106 |\n| B | D | 101 |\n| B | F | 104 |\n| B | G | 114 |\n| B | H | 103 |\n| B | J | 107 |\n| B | K | 104 |\n| B | M | 105 |\n| B | O | 102 |\n| B | Q | 111 |\n| B | S | 122 |\n| B | T | 104 |\n| B | V | 123 |\n| C | E | 108 |\n| C | G | 109 |\n| C | H | 110 |\n| C | I | 114 |\n| C | K | 122 |\n| C | L | 103 |\n| C | Q | 120 |\n| C | S | 117 |\n| C | U | 106 |\n| C | W | 102 |\n| C | Y | 117 |\n| D | E | 109 |\n| D | F | 120 |\n| D | H | 117 |\n| D | J | 107 |\n| D | K | 101 |\n| D | M | 111 |\n| D | Q | 116 |\n| D | R | 107 |\n| D | W | 114 |\n| E | L | 120 |\n| E | N | 119 |\n| E | P | 105 |\n| E | Q | 104 |\n| E | R | 118 |\n| E | W | 122 |\n| E | X | 103 |\n| F | H | 107 |\n| F | K | 115 |\n| F | L | 122 |\n| F | N | 117 |\n| F | S | 103 |\n| F | U | 115 |\n| F | V | 122 |\n| F | W | 106 |\n| G | H | 123 |\n| G | I | 112 |\n| G | K | 105 |\n| G | L | 120 |\n| G | O | 116 |\n| G | Q | 109 |\n| G | R | 119 |\n| G | S | 104 |\n| G | W | 101 |\n| G | X | 123 |\n| H | K | 109 |\n| H | O | 104 |\n| H | Q | 118 |\n| H | T | 110 |\n| H | U | 111 |\n| H | V | 110 |\n| H | Y | 119 |\n| I | M | 102 |\n| I | O | 120 |\n| I | P | 113 |\n| I | R | 124 |\n| I | S | 101 |\n| I | T | 110 |\n| I | V | 108 |\n| I | W | 122 |\n| I | Y | 106 |\n| J | R | 116 |\n| J | V | 105 |\n| J | W | 107 |\n| J | Y | 105 |\n| K | L | 107 |\n| K | M | 121 |\n| K | N | 118 |\n| K | O | 123 |\n| K | S | 102 |\n| K | U | 105 |\n| K | W | 112 |\n| K | X | 105 |\n| K | Y | 110 |\n| L | M | 114 |\n| L | O | 111 |\n| L | P | 104 |\n| L | R | 124 |\n| L | T | 112 |\n| L | U | 107 |\n| L | W | 124 |\n| M | N | 110 |\n| M | O | 117 |\n| M | P | 107 |\n| M | S | 117 |\n| M | V | 123 |\n| N | O | 113 |\n| N | U | 110 |\n| N | V | 112 |\n| N | W | 104 |\n| N | X | 103 |\n| N | Y | 123 |\n| O | P | 105 |\n| O | U | 109 |\n| O | V | 124 |\n| O | W | 110 |\n| O | X | 101 |\n| O | Y | 104 |\n| P | R | 104 |\n| P | S | 106 |\n| P | Y | 119 |\n| Q | U | 100 |\n| Q | V | 121 |\n| Q | X | 113 |\n| Q | Y | 105 |\n| R | T | 113 |\n| R | U | 117 |\n| R | W | 102 |\n| R | X | 112 |\n| S | T | 109 |\n| S | U | 101 |\n| S | W | 102 |\n| T | U | 114 |\n| T | X | 110 |\n| U | V | 103 |\n| U | W | 105 |\n| V | W | 110 |\n| V | Y | 114 |\n\nThe volunteers used these tag details to pick the combination that fits the bed and yields the greatest total beauty.\n\nIf you want, just drop your chosen plants into a tiny JSON snippet like the one below so it's easy to read and verify.\n\n{\n  \"solution\": [<selected_plant_id>, <selected_plant_id>, ...]\n}\n\nThis little block means: put the list of plant tag identifiers you picked under \"solution\". Think of each placeholder inside the brackets as the exact tag you’ll choose — replace each <selected_plant_id> with a real tag from the instance. It’s just a sketch of the shape I need, not your final answer.\n\nPlease make sure to use the identifiers exactly as they appear in the instance input — no renaming and no new labels.\n- for example: \"Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.\"",
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  {
    "task_name": "QKP",
    "problem_type": "QKP",
    "instruction": "Many photographers know this scenario: a portrait job, a single bag, and a pile of lenses and accessories to choose from. Each piece carries weight and brings a base creative benefit, and some pairs multiply the possibilities when they’re together. The point is to decide which items to bring so the total weight remains under the bag’s capacity while the total creative score—calculated by summing each packed item’s value and adding any extra pair rewards—is as large as possible; every item is optional but indivisible. The exact kit list and stats appear below.\n\nThere are 25 distinct pieces listed as 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, and the bag's weight limit is 1926 — choose which to pack so the kit fits and the creative score is maximized.\n\n| gear_id | base_creative_value |\n|---|---|\n| 1 | 100 |\n| 2 | 105 |\n| 3 | 104 |\n| 4 | 104 |\n| 5 | 101 |\n| 6 | 101 |\n| 7 | 103 |\n| 8 | 105 |\n| 9 | 100 |\n| 10 | 101 |\n| 11 | 102 |\n| 12 | 100 |\n| 13 | 103 |\n| 14 | 105 |\n| 15 | 104 |\n| 16 | 104 |\n| 17 | 102 |\n| 18 | 102 |\n| 19 | 100 |\n| 20 | 101 |\n| 21 | 103 |\n| 22 | 103 |\n| 23 | 105 |\n| 24 | 103 |\n| 25 | 102 |\n\n| gear_id | gear_weight |\n|---|---|\n| 1 | 100 |\n| 2 | 102 |\n| 3 | 101 |\n| 4 | 103 |\n| 5 | 102 |\n| 6 | 105 |\n| 7 | 104 |\n| 8 | 101 |\n| 9 | 103 |\n| 10 | 103 |\n| 11 | 104 |\n| 12 | 105 |\n| 13 | 100 |\n| 14 | 103 |\n| 15 | 100 |\n| 16 | 102 |\n| 17 | 105 |\n| 18 | 100 |\n| 19 | 100 |\n| 20 | 104 |\n| 21 | 100 |\n| 22 | 100 |\n| 23 | 103 |\n| 24 | 103 |\n| 25 | 102 |\n\n| gear_i_id | gear_j_id | pair_synergy_bonus |\n|---|---|---|\n| 1 | 3 | 100 |\n| 1 | 6 | 100 |\n| 1 | 7 | 100 |\n| 1 | 10 | 100 |\n| 1 | 11 | 100 |\n| 1 | 13 | 100 |\n| 1 | 16 | 100 |\n| 1 | 17 | 100 |\n| 1 | 19 | 100 |\n| 1 | 20 | 100 |\n| 1 | 23 | 100 |\n| 1 | 24 | 100 |\n| 1 | 25 | 100 |\n| 2 | 5 | 102 |\n| 2 | 8 | 102 |\n| 2 | 10 | 102 |\n| 2 | 11 | 102 |\n| 2 | 12 | 102 |\n| 2 | 13 | 102 |\n| 2 | 14 | 102 |\n| 2 | 17 | 100 |\n| 2 | 20 | 101 |\n| 2 | 22 | 101 |\n| 2 | 23 | 102 |\n| 2 | 25 | 100 |\n| 3 | 5 | 101 |\n| 3 | 6 | 101 |\n| 3 | 8 | 101 |\n| 3 | 13 | 100 |\n| 3 | 14 | 101 |\n| 3 | 16 | 100 |\n| 3 | 17 | 101 |\n| 3 | 20 | 101 |\n| 3 | 21 | 101 |\n| 3 | 22 | 101 |\n| 3 | 23 | 101 |\n| 3 | 24 | 100 |\n| 4 | 5 | 103 |\n| 4 | 6 | 103 |\n| 4 | 7 | 103 |\n| 4 | 8 | 103 |\n| 4 | 10 | 103 |\n| 4 | 12 | 100 |\n| 4 | 14 | 101 |\n| 4 | 18 | 103 |\n| 4 | 20 | 102 |\n| 4 | 21 | 103 |\n| 4 | 22 | 102 |\n| 4 | 24 | 102 |\n| 5 | 6 | 102 |\n| 5 | 9 | 102 |\n| 5 | 14 | 102 |\n| 5 | 15 | 102 |\n| 5 | 17 | 100 |\n| 5 | 20 | 100 |\n| 5 | 21 | 100 |\n| 5 | 22 | 100 |\n| 5 | 24 | 100 |\n| 6 | 9 | 101 |\n| 6 | 11 | 101 |\n| 6 | 12 | 105 |\n| 6 | 15 | 101 |\n| 6 | 16 | 101 |\n| 6 | 17 | 101 |\n| 6 | 18 | 104 |\n| 6 | 21 | 103 |\n| 6 | 22 | 103 |\n| 6 | 24 | 103 |\n| 6 | 25 | 102 |\n| 7 | 9 | 102 |\n| 7 | 11 | 100 |\n| 7 | 12 | 101 |\n| 7 | 15 | 104 |\n| 7 | 16 | 103 |\n| 7 | 17 | 101 |\n| 7 | 18 | 104 |\n| 7 | 20 | 103 |\n| 7 | 21 | 101 |\n| 7 | 23 | 101 |\n| 8 | 9 | 101 |\n| 8 | 12 | 101 |\n| 8 | 13 | 101 |\n| 8 | 14 | 101 |\n| 8 | 18 | 101 |\n| 8 | 19 | 104 |\n| 8 | 20 | 104 |\n| 8 | 21 | 100 |\n| 8 | 23 | 104 |\n| 8 | 24 | 100 |\n| 8 | 25 | 104 |\n| 9 | 12 | 101 |\n| 9 | 14 | 100 |\n| 9 | 15 | 102 |\n| 9 | 16 | 103 |\n| 9 | 17 | 103 |\n| 9 | 19 | 100 |\n| 9 | 24 | 100 |\n| 9 | 25 | 100 |\n| 10 | 11 | 103 |\n| 10 | 13 | 101 |\n| 10 | 15 | 102 |\n| 10 | 16 | 103 |\n| 10 | 18 | 100 |\n| 10 | 22 | 100 |\n| 10 | 23 | 100 |\n| 11 | 14 | 104 |\n| 11 | 16 | 102 |\n| 11 | 17 | 104 |\n| 11 | 18 | 104 |\n| 11 | 20 | 101 |\n| 11 | 22 | 104 |\n| 11 | 25 | 102 |\n| 12 | 15 | 104 |\n| 12 | 20 | 100 |\n| 12 | 21 | 100 |\n| 12 | 23 | 100 |\n| 12 | 24 | 100 |\n| 12 | 25 | 100 |\n| 13 | 16 | 103 |\n| 13 | 17 | 104 |\n| 13 | 20 | 102 |\n| 13 | 21 | 100 |\n| 13 | 22 | 104 |\n| 13 | 25 | 104 |\n| 14 | 15 | 100 |\n| 14 | 18 | 100 |\n| 14 | 21 | 100 |\n| 14 | 22 | 100 |\n| 14 | 23 | 100 |\n| 14 | 24 | 100 |\n| 15 | 18 | 101 |\n| 15 | 19 | 104 |\n| 15 | 20 | 100 |\n| 15 | 22 | 104 |\n| 15 | 24 | 101 |\n| 16 | 21 | 103 |\n| 16 | 22 | 103 |\n| 16 | 23 | 100 |\n| 17 | 18 | 104 |\n| 17 | 19 | 104 |\n| 17 | 23 | 100 |\n| 17 | 24 | 100 |\n| 17 | 25 | 100 |\n| 18 | 19 | 101 |\n| 18 | 20 | 101 |\n| 19 | 23 | 103 |\n| 19 | 24 | 103 |\n| 20 | 21 | 101 |\n| 20 | 22 | 101 |\n| 20 | 23 | 100 |\n| 20 | 24 | 100 |\n| 20 | 25 | 100 |\n| 21 | 23 | 103 |\n| 21 | 25 | 101 |\n| 23 | 25 | 102 |\n| 24 | 25 | 102 |\n\nPack wisely to maximize creative value without exceeding the 1926 limit.\n\nIf you want to hand me your kit choice, just drop it in a tiny JSON snippet like this:\n\n{\n  \"solution\": [<selected_item_id>, <selected_item_id>, ...]\n}\n\nHere \"solution\" is just a list of the items you’re packing — each entry should be the exact item identifier from the instance (the ID itself, not a description). Think of it like the short checklist of lenses and accessories you’ll toss into the bag. This JSON is only a sketch of the shape I expect, not the actual answer.\n\nAll identifiers must be used exactly as they appear in the instance input — no renaming and no new labels.\nValid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.",
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      "obj": 11279.0,
      "problem_type": "QKP"
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    "solution": [
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      "problem_type": "QKP",
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      "capacity": 1926,
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          "linear_profit": 105
        },
        {
          "item_id": 3,
          "linear_profit": 104
        },
        {
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          "linear_profit": 104
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          "linear_profit": 100
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          "linear_profit": 101
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          "weight": 102
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          "weight": 105
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          "weight": 100
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          "weight": 103
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          "weight": 102
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      "quadratic_pairs": [
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          "item_j_id": 1,
          "quadratic_profit": 100
        },
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          "item_j_id": 3,
          "quadratic_profit": 100
        },
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          "quadratic_profit": 100
        },
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          "item_j_id": 10,
          "quadratic_profit": 100
        },
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          "item_i_id": 1,
          "item_j_id": 11,
          "quadratic_profit": 100
        },
        {
          "item_i_id": 1,
          "item_j_id": 13,
          "quadratic_profit": 100
        },
        {
          "item_i_id": 1,
          "item_j_id": 16,
          "quadratic_profit": 100
        },
        {
          "item_i_id": 1,
          "item_j_id": 17,
          "quadratic_profit": 100
        },
        {
          "item_i_id": 1,
          "item_j_id": 19,
          "quadratic_profit": 100
        },
        {
          "item_i_id": 1,
          "item_j_id": 20,
          "quadratic_profit": 100
        },
        {
          "item_i_id": 1,
          "item_j_id": 23,
          "quadratic_profit": 100
        },
        {
          "item_i_id": 1,
          "item_j_id": 24,
          "quadratic_profit": 100
        },
        {
          "item_i_id": 1,
          "item_j_id": 25,
          "quadratic_profit": 100
        },
        {
          "item_i_id": 2,
          "item_j_id": 2,
          "quadratic_profit": 102
        },
        {
          "item_i_id": 2,
          "item_j_id": 5,
          "quadratic_profit": 102
        },
        {
          "item_i_id": 2,
          "item_j_id": 8,
          "quadratic_profit": 102
        },
        {
          "item_i_id": 2,
          "item_j_id": 10,
          "quadratic_profit": 102
        },
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    "instruction": "Many product teams face this: a fixed pot of money, a list of features each with a cost and its own benefit, and a few feature pairs that create extra value when shipped together. The challenge is to choose which features to fund so total spending stays within the pot, and the success metric is the total benefit — simply the sum of chosen features’ individual values plus any bonuses for chosen pairs — while never funding a feature twice or in parts. The exact numbers for costs, standalone benefits, and pair synergies are provided below.\n\n# num_features=25\n# total_budget=2915\n# feature_ids=A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V, W, X, Y\n\nfeature_id,standalone_value\nA,139\nB,139\nC,148\nD,136\nE,164\nF,145\nG,147\nH,160\nI,145\nJ,124\nK,150\nL,150\nM,141\nN,147\nO,129\nP,145\nQ,155\nR,121\nS,137\nT,151\nU,140\nV,156\nW,150\nX,139\nY,130\n\nfeature_id,cost\nA,124\nB,124\nC,133\nD,121\nE,149\nF,130\nG,132\nH,145\nI,130\nJ,109\nK,135\nL,135\nM,126\nN,132\nO,114\nP,130\nQ,140\nR,106\nS,122\nT,136\nU,125\nV,141\nW,135\nX,124\nY,115\n\nfeature_i_id,feature_j_id,pair_synergy_value\nA,B,128\nA,C,127\nA,D,122\nA,E,124\nA,F,122\nA,G,121\nA,H,126\nA,I,116\nA,J,134\nA,K,119\nA,L,124\nA,M,129\nA,N,117\nA,O,122\nA,P,134\nA,Q,135\nA,R,118\nA,S,113\nA,T,117\nA,U,131\nA,V,135\nA,W,123\nA,X,119\nA,Y,131\nB,C,116\nB,D,132\nB,E,126\nB,F,118\nB,G,123\nB,H,126\nB,I,111\nB,J,133\nB,K,113\nB,L,125\nB,M,111\nB,N,128\nB,O,125\nB,P,135\nB,Q,131\nB,R,113\nB,S,126\nB,T,132\nB,U,124\nB,V,116\nB,W,124\nB,X,103\nB,Y,117\nC,D,119\nC,E,127\nC,F,138\nC,G,117\nC,H,115\nC,I,138\nC,J,120\nC,K,138\nC,L,135\nC,M,127\nC,N,124\nC,O,115\nC,P,125\nC,Q,133\nC,R,119\nC,S,140\nC,T,139\nC,U,116\nC,V,118\nC,W,112\nC,X,121\nC,Y,108\nD,E,121\nD,F,126\nD,G,123\nD,H,117\nD,I,132\nD,J,134\nD,K,128\nD,L,118\nD,M,121\nD,N,116\nD,O,113\nD,P,116\nD,Q,114\nD,R,125\nD,S,122\nD,T,113\nD,U,128\nD,V,134\nD,W,134\nD,X,146\nD,Y,134\nE,F,137\nE,G,134\nE,H,138\nE,I,124\nE,J,146\nE,K,130\nE,L,139\nE,M,141\nE,N,143\nE,O,123\nE,P,130\nE,Q,123\nE,R,128\nE,S,144\nE,T,135\nE,U,127\nE,V,125\nE,W,135\nE,X,125\nE,Y,126\nF,G,130\nF,H,135\nF,I,127\nF,J,115\nF,K,125\nF,L,125\nF,M,119\nF,N,125\nF,O,123\nF,P,135\nF,Q,124\nF,R,129\nF,S,127\nF,T,131\nF,U,125\nF,V,136\nF,W,122\nF,X,127\nF,Y,129\nG,H,137\nG,I,120\nG,J,115\nG,K,136\nG,L,132\nG,M,135\nG,N,118\nG,O,134\nG,P,122\nG,Q,133\nG,R,115\nG,S,133\nG,T,123\nG,U,122\nG,V,113\nG,W,125\nG,X,120\nG,Y,122\nH,I,130\nH,J,135\nH,K,135\nH,L,141\nH,M,140\nH,N,129\nH,O,145\nH,P,134\nH,Q,121\nH,R,145\nH,S,113\nH,T,113\nH,U,102\nH,V,109\nH,W,107\nH,X,102\nH,Y,102\nI,J,132\nI,K,130\nI,L,118\nI,M,124\nI,N,129\nI,O,135\nI,P,134\nI,Q,126\nI,R,113\nI,S,108\nI,T,110\nI,U,115\nI,V,109\nI,W,116\nI,X,111\nI,Y,121\nJ,K,109\nJ,L,112\nJ,M,109\nJ,N,109\nJ,O,126\nJ,P,120\nJ,Q,114\nJ,R,108\nJ,S,108\nJ,T,123\nJ,U,120\nJ,V,115\nJ,W,103\nJ,X,117\nJ,Y,103\nK,L,132\nK,M,131\nK,N,129\nK,O,129\nK,P,115\nK,Q,109\nK,R,102\nK,S,109\nK,T,114\nK,U,115\nK,V,122\nK,W,116\nK,X,113\nK,Y,107\nL,M,119\nL,N,132\nL,O,120\nL,P,122\nL,Q,131\nL,R,116\nL,S,116\nL,T,116\nL,U,119\nL,V,135\nL,W,112\nL,X,127\nL,Y,112\nM,N,100\nM,O,109\nM,P,117\nM,Q,106\nM,R,104\nM,S,114\nM,T,121\nM,U,116\nM,V,111\nM,W,102\nM,X,113\nM,Y,104\nN,O,122\nN,P,112\nN,Q,104\nN,R,121\nN,S,116\nN,T,113\nN,U,125\nN,V,132\nN,W,120\nN,X,122\nN,Y,118\nO,P,139\nO,Q,131\nO,R,128\nO,S,111\nO,T,120\nO,U,121\nO,V,129\nO,W,126\nO,X,111\nO,Y,132\nP,Q,127\nP,R,135\nP,S,137\nP,T,134\nP,U,115\nP,V,120\nP,W,136\nP,X,124\nP,Y,104\nQ,R,104\nQ,S,104\nQ,T,104\nQ,U,107\nQ,V,120\nQ,W,142\nQ,X,128\nQ,Y,130\nR,S,136\nR,T,136\nR,U,120\nR,V,130\nR,W,125\nR,X,128\nR,Y,125\nS,T,113\nS,U,114\nS,V,110\nS,W,129\nS,X,106\nS,Y,128\nT,U,134\nT,V,135\nT,W,125\nT,X,132\nT,Y,114\nU,V,119\nU,W,118\n\nWhen you're ready to give me the selection, just use this simple JSON shape so I can read it automatically — nothing fancy, just a list of the item IDs you want to pick:\n\n{\n  \"solution\": [<selected_item_id>, <selected_item_id>, ...]\n}\n\n\"solution\" is just the list of features/items you choose to fund. Think of it like filling out a form: put each item's ID in the array, and that's your selection. This block is just a sketch of the expected shape — not the real answer itself.\n\nPlease make sure to use the exact identifiers from the instance input — don't rename them or invent new labels. \nValid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.",
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      "C",
      "D",
      "E",
      "F",
      "G",
      "H",
      "I",
      "J",
      "K",
      "L",
      "M",
      "N",
      "O",
      "P",
      "Q",
      "R",
      "S",
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  {
    "task_name": "QKP",
    "problem_type": "QKP",
    "instruction": "We were putting together a film lineup for a friends’ marathon: every movie brings its own entertainment value and takes a chunk of the evening, and certain pairings boost the fun when both are included. The plan was to add up the happiness from each selected movie plus any extra enjoyment from pairs, while keeping the total runtime within the night’s limit and not repeating films, so the best lineup is the one with the largest total enjoyment that still fits. The specific runtimes and pairing bonuses are listed below.\n\nWe had 22 films to pick from (A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V) and 37401 minutes for the marathon.\nFor film A we get 3231 base enjoyment on its own.\nFor film B we get 537 base enjoyment on its own.\nFor film C we get 2825 base enjoyment on its own.\nFor film D we get 2294 base enjoyment on its own.\nFor film E we get 1515 base enjoyment on its own.\nFor film F we get 1279 base enjoyment on its own.\nFor film G we get 3057 base enjoyment on its own.\nFor film H we get 1409 base enjoyment on its own.\nFor film I we get 1352 base enjoyment on its own.\nFor film J we get 1060 base enjoyment on its own.\nFor film K we get 2085 base enjoyment on its own.\nFor film L we get 298 base enjoyment on its own.\nFor film M we get 1341 base enjoyment on its own.\nFor film N we get 2157 base enjoyment on its own.\nFor film O we get 1165 base enjoyment on its own.\nFor film P we get 673 base enjoyment on its own.\nFor film Q we get 1151 base enjoyment on its own.\nFor film R we get 977 base enjoyment on its own.\nFor film S we get 148 base enjoyment on its own.\nFor film T we get 2208 base enjoyment on its own.\nFor film U we get 3108 base enjoyment on its own.\nFor film V we get 2094 base enjoyment on its own.\nFilm A uses up 2226 minutes of the evening.\nFilm B uses up 1123 minutes of the evening.\nFilm C uses up 2092 minutes of the evening.\nFilm D uses up 572 minutes of the evening.\nFilm E uses up 362 minutes of the evening.\nFilm F uses up 1332 minutes of the evening.\nFilm G uses up 1193 minutes of the evening.\nFilm H uses up 2301 minutes of the evening.\nFilm I uses up 2227 minutes of the evening.\nFilm J uses up 759 minutes of the evening.\nFilm K uses up 2537 minutes of the evening.\nFilm L uses up 2416 minutes of the evening.\nFilm M uses up 957 minutes of the evening.\nFilm N uses up 946 minutes of the evening.\nFilm O uses up 968 minutes of the evening.\nFilm P uses up 1522 minutes of the evening.\nFilm Q uses up 3080 minutes of the evening.\nFilm R uses up 3197 minutes of the evening.\nFilm S uses up 1992 minutes of the evening.\nFilm T uses up 3002 minutes of the evening.\nFilm U uses up 2211 minutes of the evening.\nFilm V uses up 2327 minutes of the evening.\nIf we include A and B together, they add 1853 extra enjoyment.\nIf we include A and C together, they add 994 extra enjoyment.\nIf we include A and G together, they add 2471 extra enjoyment.\nIf we include A and P together, they add 2512 extra enjoyment.\nIf we include A and R together, they add 2105 extra enjoyment.\nIf we include A and T together, they add 2555 extra enjoyment.\nIf we include A and V together, they add 813 extra enjoyment.\nIf we include B and G together, they add 1278 extra enjoyment.\nIf we include B and I together, they add 891 extra enjoyment.\nIf we include B and K together, they add 1385 extra enjoyment.\nIf we include B and O together, they add 1850 extra enjoyment.\nIf we include C and D together, they add 1571 extra enjoyment.\nIf we include C and P together, they add 2586 extra enjoyment.\nIf we include D and K together, they add 995 extra enjoyment.\nIf we include D and S together, they add 842 extra enjoyment.\nIf we include D and V together, they add 1845 extra enjoyment.\nIf we include E and H together, they add 538 extra enjoyment.\nIf we include E and J together, they add 917 extra enjoyment.\nIf we include E and K together, they add 217 extra enjoyment.\nIf we include E and O together, they add 901 extra enjoyment.\nIf we include E and S together, they add 1581 extra enjoyment.\nIf we include F and T together, they add 2208 extra enjoyment.\nIf we include G and I together, they add 977 extra enjoyment.\nIf we include G and N together, they add 1621 extra enjoyment.\nIf we include G and Q together, they add 2833 extra enjoyment.\nIf we include G and U together, they add 1734 extra enjoyment.\nIf we include H and K together, they add 1672 extra enjoyment.\nIf we include H and N together, they add 2398 extra enjoyment.\nIf we include H and O together, they add 2019 extra enjoyment.\nIf we include H and S together, they add 3088 extra enjoyment.\nIf we include I and P together, they add 766 extra enjoyment.\nIf we include J and R together, they add 1398 extra enjoyment.\nIf we include J and V together, they add 2530 extra enjoyment.\nIf we include K and N together, they add 2032 extra enjoyment.\nIf we include K and P together, they add 2805 extra enjoyment.\nIf we include K and Q together, they add 2400 extra enjoyment.\nIf we include L and M together, they add 1359 extra enjoyment.\nIf we include L and Q together, they add 1692 extra enjoyment.\nIf we include M and N together, they add 1234 extra enjoyment.\nIf we include M and P together, they add 1476 extra enjoyment.\nIf we include M and Q together, they add 1165 extra enjoyment.\nIf we include M and R together, they add 1494 extra enjoyment.\nIf we include N and P together, they add 570 extra enjoyment.\nIf we include N and T together, they add 684 extra enjoyment.\nIf we include N and U together, they add 1769 extra enjoyment.\nIf we include N and V together, they add 1689 extra enjoyment.\nIf we include P and V together, they add 2576 extra enjoyment.\nIf we include Q and R together, they add 1862 extra enjoyment.\nIf we include R and U together, they add 2103 extra enjoyment.\nNow we can list each film's enjoyment, runtime, and pair bonuses so we pick the lineup with the largest total enjoyment that still fits into 37401 minutes.\n\nAlso, when you send back your chosen lineup, a tiny JSON snippet like the one below works great — simple and machine-friendly.\n\n{\n  \"solution\": [<selected_movie_id>, <selected_movie_id>, ...]\n}\n\nThis just means: \"solution\" is the list of movies you're picking for the marathon, and each <selected_movie_id> is a placeholder for one movie's identifier from the problem instance. Think of it like filling in a short form — put each movie's ID in the array (order doesn't matter, and don't repeat a movie).\n\nThe JSON above is just a sketch of the expected shape, not the actual answer — replace those angle-bracket placeholders with the real identifiers from the instance.\n\nPlease use the identifiers exactly as they appear in the instance input — no renaming, no made-up labels.\n- for example: \"Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.\"",
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        537,
        2825,
        2294,
        1515,
        1279,
        3057,
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        1352,
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        2085,
        298,
        1341,
        2157,
        1165,
        673,
        1151,
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          994,
          0,
          0,
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The concrete details follow below.\n\nWe have 20 pastries (A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T) and the box capacity is 1102.\n\n| pastry_id | individual_appeal |\n|---|---|\n| A | 105 |\n| B | 104 |\n| C | 100 |\n| D | 100 |\n| E | 104 |\n| F | 104 |\n| G | 101 |\n| H | 103 |\n| I | 102 |\n| J | 100 |\n| K | 101 |\n| L | 101 |\n| M | 100 |\n| N | 104 |\n| O | 105 |\n| P | 103 |\n| Q | 100 |\n| R | 102 |\n| S | 105 |\n| T | 103 |\n\n| pastry_id | pastry_weight |\n|---|---|\n| A | 101 |\n| B | 104 |\n| C | 105 |\n| D | 101 |\n| E | 101 |\n| F | 101 |\n| G | 104 |\n| H | 104 |\n| I | 102 |\n| J | 105 |\n| K | 101 |\n| L | 105 |\n| M | 102 |\n| N | 105 |\n| O | 102 |\n| P | 103 |\n| Q | 104 |\n| R | 100 |\n| S | 104 |\n| T | 101 |\n\n| pastry_i_id | pastry_j_id | pair_bonus_appeal |\n|---|---|---|\n| A | B | 101 |\n| A | C | 100 |\n| A | D | 101 |\n| A | E | 101 |\n| A | F | 101 |\n| A | G | 101 |\n| A | H | 100 |\n| A | I | 100 |\n| A | J | 101 |\n| A | K | 101 |\n| A | L | 100 |\n| A | M | 101 |\n| A | N | 101 |\n| A | O | 102 |\n| A | P | 102 |\n| A | Q | 101 |\n| A | R | 100 |\n| A | S | 102 |\n| A | T | 101 |\n| B | C | 103 |\n| B | D | 102 |\n| B | E | 102 |\n| B | F | 102 |\n| B | G | 101 |\n| B | H | 104 |\n| B | I | 104 |\n| B | J | 104 |\n| B | K | 104 |\n| B | L | 101 |\n| B | M | 104 |\n| B | N | 102 |\n| B | O | 102 |\n| B | P | 101 |\n| B | Q | 104 |\n| B | R | 104 |\n| B | S | 101 |\n| B | T | 102 |\n| C | D | 102 |\n| C | E | 102 |\n| C | F | 103 |\n| C | G | 103 |\n| C | H | 104 |\n| C | I | 104 |\n| C | J | 105 |\n| C | K | 102 |\n| C | L | 105 |\n| C | M | 103 |\n| C | N | 103 |\n| C | O | 105 |\n| C | P | 103 |\n| C | Q | 103 |\n| C | R | 102 |\n| C | S | 100 |\n| C | T | 101 |\n| D | E | 101 |\n| D | F | 101 |\n| D | G | 100 |\n| D | H | 101 |\n| D | I | 102 |\n| D | J | 102 |\n| D | K | 101 |\n| D | L | 101 |\n| D | M | 101 |\n| D | N | 102 |\n| D | O | 101 |\n| D | P | 101 |\n| D | Q | 102 |\n| D | R | 101 |\n| D | S | 100 |\n| D | T | 102 |\n| E | F | 101 |\n| E | G | 102 |\n| E | H | 101 |\n| E | I | 100 |\n| E | J | 102 |\n| E | K | 101 |\n| E | L | 101 |\n| E | M | 101 |\n| E | N | 102 |\n| E | O | 101 |\n| E | P | 100 |\n| E | Q | 102 |\n| E | R | 102 |\n| E | S | 102 |\n| E | T | 101 |\n| F | G | 100 |\n| F | H | 102 |\n| F | I | 100 |\n| F | J | 100 |\n| F | K | 100 |\n| F | L | 102 |\n| F | M | 102 |\n| F | N | 102 |\n| F | O | 100 |\n| F | P | 102 |\n| F | Q | 102 |\n| F | R | 104 |\n| F | S | 104 |\n| F | T | 102 |\n| G | H | 104 |\n| G | I | 104 |\n| G | J | 102 |\n| G | K | 103 |\n| G | L | 103 |\n| G | M | 102 |\n| G | N | 103 |\n| G | O | 104 |\n| G | P | 103 |\n| G | Q | 102 |\n| G | R | 104 |\n| G | S | 103 |\n| G | T | 103 |\n| H | I | 101 |\n| H | J | 101 |\n| H | K | 102 |\n| H | L | 104 |\n| H | M | 103 |\n| H | N | 102 |\n| H | O | 101 |\n| H | P | 100 |\n| H | Q | 101 |\n| H | R | 100 |\n| H | S | 100 |\n| H | T | 101 |\n| I | J | 101 |\n| I | K | 100 |\n| I | L | 101 |\n| I | M | 100 |\n| I | N | 100 |\n| I | O | 101 |\n| I | P | 101 |\n| I | Q | 100 |\n| I | R | 100 |\n| I | S | 101 |\n| I | T | 101 |\n| J | K | 102 |\n| J | L | 102 |\n| J | M | 103 |\n| J | N | 102 |\n| J | O | 103 |\n| J | P | 104 |\n| J | Q | 103 |\n| J | R | 103 |\n| J | S | 103 |\n| J | T | 105 |\n| K | L | 102 |\n| K | M | 101 |\n| K | N | 101 |\n| K | O | 101 |\n| K | P | 100 |\n| K | Q | 102 |\n| K | R | 101 |\n| K | S | 101 |\n| K | T | 100 |\n| L | M | 105 |\n| L | N | 104 |\n| L | O | 103 |\n| L | P | 103 |\n| L | Q | 101 |\n| L | R | 101 |\n| L | S | 102 |\n| L | T | 103 |\n| M | N | 101 |\n| M | O | 101 |\n| M | P | 102 |\n| M | Q | 102 |\n| M | R | 102 |\n| M | S | 101 |\n| M | T | 100 |\n| N | O | 102 |\n| N | P | 100 |\n| N | Q | 102 |\n| N | R | 103 |\n| N | S | 100 |\n| N | T | 101 |\n| O | P | 102 |\n| O | Q | 102 |\n| O | R | 103 |\n| O | S | 103 |\n| O | T | 103 |\n| P | Q | 103 |\n| P | R | 103 |\n| P | S | 102 |\n| P | T | 102 |\n| Q | R | 102 |\n| Q | S | 101 |\n| Q | T | 100 |\n| R | S | 104 |\n| R | T | 102 |\n\nWe'll pick the combination that fits within 1102 and maximizes the summed appeal.\n\nAlso, when you send back the final selection, just drop it in a tiny JSON snippet like this so it's easy to parse:\n\n{\n  \"solution\": [<selected_item_id>, <selected_item_id>, ...]\n}\n\n\"solution\" is simply the list of pastry IDs that should go into the box. Think of it as the short form of your order: just the exact IDs of the pastries you picked. This block is only a sketch of the expected shape — not the actual answer.\n\nPlease use the identifiers exactly as they appear in the instance input — no renaming and no new labels. Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.",
    "instance": {
      "linear_coeffs": [
        105,
        104,
        100,
        100,
        104,
        104,
        101,
        103,
        102,
        100,
        101,
        101,
        100,
        104,
        105,
        103,
        100,
        102,
        105,
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      ],
      "quadratic_coeffs": [
        [
          100,
          101,
          100,
          101,
          101,
          101,
          101,
          100,
          100,
          101,
          101,
          100,
          101,
          101,
          102,
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          101,
          100,
          102,
          101
        ],
        [
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          102,
          102,
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          101,
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          104,
          104,
          104,
          101,
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          101,
          104,
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          101,
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        ],
        [
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          102,
          100,
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        ],
        [
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          101,
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          102,
          101,
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        ],
        [
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        ],
        [
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        ],
        [
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        ],
        [
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        ],
        [
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        ],
        [
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        ],
        [
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        [
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        [
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        [
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        [
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        ],
        [
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      ],
      "weights": [
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        101,
        101,
        101,
        104,
        104,
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        105,
        101,
        105,
        102,
        105,
        102,
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        104,
        100,
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      ],
      "capacity": 1102,
      "solution": [
        1,
        2,
        7,
        8,
        9,
        11,
        13,
        14,
        16,
        17
      ],
      "obj": 5629.0,
      "problem_type": "QKP"
    },
    "solution": [
      1,
      2,
      7,
      8,
      9,
      11,
      13,
      14,
      16,
      17
    ],
    "obj": 5629.0,
    "instance_variant": {
      "problem_type": "QKP",
      "num_items": 20,
      "capacity": 1102,
      "items": [
        "A",
        "B",
        "C",
        "D",
        "E",
        "F",
        "G",
        "H",
        "I",
        "J",
        "K",
        "L",
        "M",
        "N",
        "O",
        "P",
        "Q",
        "R",
        "S",
        "T"
      ],
      "linear_pairs": [
        {
          "item_id": "A",
          "linear_profit": 105
        },
        {
          "item_id": "B",
          "linear_profit": 104
        },
        {
          "item_id": "C",
          "linear_profit": 100
        },
        {
          "item_id": "D",
          "linear_profit": 100
        },
        {
          "item_id": "E",
          "linear_profit": 104
        },
        {
          "item_id": "F",
          "linear_profit": 104
        },
        {
          "item_id": "G",
          "linear_profit": 101
        },
        {
          "item_id": "H",
          "linear_profit": 103
        },
        {
          "item_id": "I",
          "linear_profit": 102
        },
        {
          "item_id": "J",
          "linear_profit": 100
        },
        {
          "item_id": "K",
          "linear_profit": 101
        },
        {
          "item_id": "L",
          "linear_profit": 101
        },
        {
          "item_id": "M",
          "linear_profit": 100
        },
        {
          "item_id": "N",
          "linear_profit": 104
        },
        {
          "item_id": "O",
          "linear_profit": 105
        },
        {
          "item_id": "P",
          "linear_profit": 103
        },
        {
          "item_id": "Q",
          "linear_profit": 100
        },
        {
          "item_id": "R",
          "linear_profit": 102
        },
        {
          "item_id": "S",
          "linear_profit": 105
        },
        {
          "item_id": "T",
          "linear_profit": 103
        }
      ],
      "weight_pairs": [
        {
          "item_id": "A",
          "weight": 101
        },
        {
          "item_id": "B",
          "weight": 104
        },
        {
          "item_id": "C",
          "weight": 105
        },
        {
          "item_id": "D",
          "weight": 101
        },
        {
          "item_id": "E",
          "weight": 101
        },
        {
          "item_id": "F",
          "weight": 101
        },
        {
          "item_id": "G",
          "weight": 104
        },
        {
          "item_id": "H",
          "weight": 104
        },
        {
          "item_id": "I",
          "weight": 102
        },
        {
          "item_id": "J",
          "weight": 105
        },
        {
          "item_id": "K",
          "weight": 101
        },
        {
          "item_id": "L",
          "weight": 105
        },
        {
          "item_id": "M",
          "weight": 102
        },
        {
          "item_id": "N",
          "weight": 105
        },
        {
          "item_id": "O",
          "weight": 102
        },
        {
          "item_id": "P",
          "weight": 103
        },
        {
          "item_id": "Q",
          "weight": 104
        },
        {
          "item_id": "R",
          "weight": 100
        },
        {
          "item_id": "S",
          "weight": 104
        },
        {
          "item_id": "T",
          "weight": 101
        }
      ],
      "quadratic_pairs": [
        {
          "item_i_id": "A",
          "item_j_id": "A",
          "quadratic_profit": 100
        },
        {
          "item_i_id": "A",
          "item_j_id": "B",
          "quadratic_profit": 101
        },
        {
          "item_i_id": "A",
          "item_j_id": "C",
          "quadratic_profit": 100
        },
        {
          "item_i_id": "A",
          "item_j_id": "D",
          "quadratic_profit": 101
        },
        {
          "item_i_id": "A",
          "item_j_id": "E",
          "quadratic_profit": 101
        },
        {
          "item_i_id": "A",
          "item_j_id": "F",
          "quadratic_profit": 101
        },
        {
          "item_i_id": "A",
          "item_j_id": "G",
          "quadratic_profit": 101
        },
        {
          "item_i_id": "A",
          "item_j_id": "H",
          "quadratic_profit": 100
        },
        {
          "item_i_id": "A",
          "item_j_id": "I",
          "quadratic_profit": 100
        },
        {
          "item_i_id": "A",
          "item_j_id": "J",
          "quadratic_profit": 101
        },
        {
          "item_i_id": "A",
          "item_j_id": "K",
          "quadratic_profit": 101
        },
        {
          "item_i_id": "A",
          "item_j_id": "L",
          "quadratic_profit": 100
        },
        {
          "item_i_id": "A",
          "item_j_id": "M",
          "quadratic_profit": 101
        },
        {
          "item_i_id": "A",
          "item_j_id": "N",
          "quadratic_profit": 101
        },
        {
          "item_i_id": "A",
          "item_j_id": "O",
          "quadratic_profit": 102
        },
        {
          "item_i_id": "A",
          "item_j_id": "P",
          "quadratic_profit": 102
        },
        {
          "item_i_id": "A",
          "item_j_id": "Q",
          "quadratic_profit": 101
        },
        {
          "item_i_id": "A",
          "item_j_id": "R",
          "quadratic_profit": 100
        },
        {
          "item_i_id": "A",
          "item_j_id": "S",
          "quadratic_profit": 102
        },
        {
          "item_i_id": "A",
          "item_j_id": "T",
          "quadratic_profit": 101
        },
        {
          "item_i_id": "B",
          "item_j_id": "B",
          "quadratic_profit": 104
        },
        {
          "item_i_id": "B",
          "item_j_id": "C",
          "quadratic_profit": 103
        },
        {
          "item_i_id": "B",
          "item_j_id": "D",
          "quadratic_profit": 102
        },
        {
          "item_i_id": "B",
          "item_j_id": "E",
          "quadratic_profit": 102
        },
        {
          "item_i_id": "B",
          "item_j_id": "F",
          "quadratic_profit": 102
        },
        {
          "item_i_id": "B",
          "item_j_id": "G",
          "quadratic_profit": 101
        },
        {
          "item_i_id": "B",
          "item_j_id": "H",
          "quadratic_profit": 104
        },
        {
          "item_i_id": "B",
          "item_j_id": "I",
          "quadratic_profit": 104
        },
        {
          "item_i_id": "B",
          "item_j_id": "J",
          "quadratic_profit": 104
        },
        {
          "item_i_id": "B",
          "item_j_id": "K",
          "quadratic_profit": 104
        },
        {
          "item_i_id": "B",
          "item_j_id": "L",
          "quadratic_profit": 101
        },
        {
          "item_i_id": "B",
          "item_j_id": "M",
          "quadratic_profit": 104
        },
        {
          "item_i_id": "B",
          "item_j_id": "N",
          "quadratic_profit": 102
        },
        {
          "item_i_id": "B",
          "item_j_id": "O",
          "quadratic_profit": 102
        },
        {
          "item_i_id": "B",
          "item_j_id": "P",
          "quadratic_profit": 101
        },
        {
          "item_i_id": "B",
          "item_j_id": "Q",
          "quadratic_profit": 104
        },
        {
          "item_i_id": "B",
          "item_j_id": "R",
          "quadratic_profit": 104
        },
        {
          "item_i_id": "B",
          "item_j_id": "S",
          "quadratic_profit": 101
        },
        {
          "item_i_id": "B",
          "item_j_id": "T",
          "quadratic_profit": 102
        },
        {
          "item_i_id": "C",
          "item_j_id": "C",
          "quadratic_profit": 103
        },
        {
          "item_i_id": "C",
          "item_j_id": "D",
          "quadratic_profit": 102
        },
        {
          "item_i_id": "C",
          "item_j_id": "E",
          "quadratic_profit": 102
        },
        {
          "item_i_id": "C",
          "item_j_id": "F",
          "quadratic_profit": 103
        },
        {
          "item_i_id": "C",
          "item_j_id": "G",
          "quadratic_profit": 103
        },
        {
          "item_i_id": "C",
          "item_j_id": "H",
          "quadratic_profit": 104
        },
        {
          "item_i_id": "C",
          "item_j_id": "I",
          "quadratic_profit": 104
        },
        {
          "item_i_id": "C",
          "item_j_id": "J",
          "quadratic_profit": 105
        },
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    "instruction": "Someone in the group always packs like they’re moving house, so the rest of us try to be smarter: pick a set of essentials that won’t overload the pannier while getting the most convenience out of what’s carried. Each item contributes its own usefulness and some two-item combos add extra convenience when together, so the total value is the sum of chosen items’ usefulness plus any applicable pair bonuses. Items can’t be taken more than once and the total weight must stay under the pannier limit; the exact list of item weights, usefulness scores, pair bonuses, and capacity are shown below.\n\n# total_items_count=22\n# pannier_capacity=5849\n# item_ids_list=1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22\n\nitem_id,usefulness_score\n1,476\n2,276\n3,229\n4,244\n5,136\n6,282\n7,381\n8,465\n9,324\n10,328\n11,171\n12,245\n13,494\n14,202\n15,354\n16,231\n17,421\n18,285\n19,348\n20,112\n21,428\n22,313\n\nitem_id,item_weight\n1,477\n2,397\n3,210\n4,118\n5,101\n6,223\n7,414\n8,334\n9,419\n10,181\n11,387\n12,205\n13,487\n14,121\n15,223\n16,198\n17,144\n18,317\n19,126\n20,298\n21,291\n22,254\n\nfirst_item_id,second_item_id,complementary_bonus\n1,2,319\n1,3,419\n1,4,466\n1,5,208\n1,6,267\n1,7,148\n1,8,289\n1,9,483\n1,10,260\n1,11,209\n1,12,406\n1,13,299\n1,14,376\n1,15,193\n1,16,122\n1,17,197\n1,18,438\n1,19,207\n1,20,390\n1,21,188\n1,22,216\n2,3,491\n2,4,107\n2,5,180\n2,6,482\n2,7,351\n2,8,145\n2,9,195\n2,10,187\n2,11,216\n2,12,140\n2,13,156\n2,14,320\n2,15,265\n2,16,154\n2,17,258\n2,18,172\n2,19,453\n2,20,345\n2,21,227\n2,22,328\n3,4,171\n3,5,381\n3,6,474\n3,7,326\n3,8,462\n3,9,193\n3,10,329\n3,11,214\n3,12,205\n3,13,140\n3,14,386\n3,15,482\n3,16,215\n3,17,432\n3,18,244\n3,19,490\n3,20,110\n3,21,229\n3,22,183\n4,5,103\n4,6,399\n4,7,494\n4,8,290\n4,9,208\n4,10,111\n4,11,277\n4,12,277\n4,13,176\n4,14,193\n4,15,342\n4,16,418\n4,17,390\n4,18,258\n4,19,351\n4,20,407\n4,21,150\n4,22,472\n5,6,241\n5,7,304\n5,8,366\n5,9,303\n5,10,205\n5,11,280\n5,12,223\n5,13,436\n5,14,408\n5,15,493\n5,16,128\n5,17,221\n5,18,323\n5,19,333\n5,20,316\n5,21,147\n5,22,270\n6,7,111\n6,8,217\n6,9,180\n6,10,206\n6,11,432\n6,12,204\n6,13,339\n6,14,459\n6,15,423\n6,16,342\n6,17,158\n6,18,354\n6,19,242\n6,20,236\n6,21,441\n6,22,272\n7,8,141\n7,9,489\n7,10,450\n7,11,298\n7,12,128\n7,13,353\n7,14,125\n7,15,242\n7,16,452\n7,17,108\n7,18,297\n7,19,314\n7,20,165\n7,21,272\n7,22,218\n8,9,159\n8,10,320\n8,11,448\n8,12,384\n8,13,367\n8,14,257\n8,15,114\n8,16,120\n8,17,172\n8,18,498\n8,19,267\n8,20,208\n8,21,102\n8,22,223\n9,10,332\n9,11,137\n9,12,356\n9,13,351\n9,14,149\n9,15,134\n9,16,461\n9,17,411\n9,18,289\n9,19,145\n9,20,316\n9,21,366\n9,22,213\n10,11,190\n10,12,256\n10,13,177\n10,14,372\n10,15,307\n10,16,139\n10,17,251\n10,18,383\n10,19,235\n10,20,459\n10,21,156\n10,22,148\n11,12,255\n11,13,145\n11,14,166\n11,15,111\n11,16,187\n11,17,457\n11,18,116\n11,19,333\n11,20,163\n11,21,457\n11,22,446\n12,13,180\n12,14,306\n12,15,230\n12,16,421\n12,17,166\n12,18,187\n12,19,207\n12,20,494\n12,21,373\n12,22,301\n13,14,480\n13,15,438\n13,16,280\n13,17,271\n13,18,477\n13,19,128\n13,20,317\n13,21,320\n13,22,140\n14,15,191\n14,16,299\n14,17,190\n14,18,228\n14,19,174\n14,20,382\n14,21,463\n14,22,330\n15,16,458\n15,17,128\n15,18,180\n15,19,425\n15,20,329\n15,21,301\n15,22,485\n16,17,173\n16,18,421\n16,19,283\n16,20,206\n16,21,304\n16,22,250\n17,18,324\n17,19,166\n17,20,487\n17,21,370\n17,22,110\n18,19,400\n18,20,348\n18,21,242\n18,22,474\n19,20,273\n\nOh, and when you send back which items to take, a tiny heads-up: please use this simple JSON layout so it's easy to read and parse.\n\n{\n  \"solution\": [<selected_item_id>, <selected_item_id>, ...]\n}\n\nThis just means \"solution\" should be a list containing the IDs of the items you want to pack. Think of it like ticking boxes on a checklist — each entry is the exact ID of an item you choose. The JSON above is just a sketch of the shape I expect, not the actual answer.\n\nPlease use the item identifiers exactly as they appear in the instance input — don’t rename them or invent new labels. Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.",
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    "task_name": "QKP",
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    "instruction": "Someone on the ops team had to trim features for a constrained machine: every feature takes up memory and brings a baseline benefit, and a few feature pairings produce extra value if both are turned on. The job was to decide which features to switch on so the total memory they require doesn’t exceed what’s free, and the overall system value is as high as it can be — that value is just the sum of the selected features’ base benefits plus any bonuses from feature pairs that are active together. Features are either active or inactive (no duplicates), and the specific memory costs and benefit numbers appear below.\n\nThere were 19 optional modules listed as 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, and 15677 memory units were available for enabling them.\n\n| module_id | base_benefit |\n|---|---|\n| 1 | 713 |\n| 2 | 625 |\n| 3 | 672 |\n| 4 | 1415 |\n| 5 | 1290 |\n| 6 | 178 |\n| 7 | 763 |\n| 8 | 827 |\n| 9 | 1214 |\n| 10 | 274 |\n| 11 | 1672 |\n| 12 | 1460 |\n| 13 | 235 |\n| 14 | 1324 |\n| 15 | 1054 |\n| 16 | 408 |\n| 17 | 733 |\n| 18 | 205 |\n| 19 | 772 |\n\n| module_id | memory_usage |\n|---|---|\n| 1 | 413 |\n| 2 | 1019 |\n| 3 | 603 |\n| 4 | 962 |\n| 5 | 1480 |\n| 6 | 1508 |\n| 7 | 536 |\n| 8 | 941 |\n| 9 | 1254 |\n| 10 | 1295 |\n| 11 | 1441 |\n| 12 | 667 |\n| 13 | 823 |\n| 14 | 1414 |\n| 15 | 1161 |\n| 16 | 920 |\n| 17 | 1057 |\n| 18 | 1434 |\n| 19 | 1113 |\n\n| module_i_id | module_j_id | interaction_bonus |\n|---|---|---|\n| 1 | 3 | 413 |\n| 1 | 4 | 413 |\n| 1 | 5 | 413 |\n| 1 | 6 | 413 |\n| 1 | 7 | 413 |\n| 1 | 8 | 413 |\n| 1 | 10 | 413 |\n| 1 | 11 | 139 |\n| 1 | 14 | 413 |\n| 1 | 16 | 413 |\n| 1 | 17 | 152 |\n| 1 | 18 | 413 |\n| 1 | 19 | 500 |\n| 2 | 3 | 834 |\n| 2 | 5 | 1019 |\n| 2 | 7 | 1019 |\n| 2 | 8 | 440 |\n| 2 | 9 | 1019 |\n| 2 | 10 | 139 |\n| 2 | 13 | 1019 |\n| 2 | 14 | 1019 |\n| 2 | 15 | 920 |\n| 2 | 16 | 690 |\n| 2 | 17 | 861 |\n| 2 | 19 | 1199 |\n| 3 | 4 | 603 |\n| 3 | 7 | 603 |\n| 3 | 9 | 603 |\n| 3 | 10 | 603 |\n| 3 | 12 | 603 |\n| 3 | 13 | 603 |\n| 3 | 18 | 318 |\n| 4 | 7 | 182 |\n| 4 | 8 | 551 |\n| 4 | 10 | 962 |\n| 4 | 11 | 962 |\n| 4 | 13 | 962 |\n| 4 | 14 | 962 |\n| 4 | 15 | 962 |\n| 4 | 16 | 1435 |\n| 4 | 17 | 941 |\n| 4 | 18 | 1435 |\n| 5 | 6 | 323 |\n| 5 | 7 | 902 |\n| 5 | 9 | 675 |\n| 5 | 11 | 1478 |\n| 5 | 12 | 1048 |\n| 5 | 14 | 920 |\n| 5 | 15 | 690 |\n| 5 | 16 | 1508 |\n| 5 | 17 | 1508 |\n| 5 | 18 | 771 |\n| 5 | 19 | 323 |\n| 6 | 7 | 1508 |\n| 6 | 8 | 408 |\n| 6 | 13 | 998 |\n| 6 | 15 | 152 |\n| 6 | 18 | 536 |\n| 7 | 9 | 536 |\n| 7 | 10 | 245 |\n| 7 | 11 | 154 |\n| 7 | 13 | 536 |\n| 7 | 15 | 536 |\n| 7 | 16 | 834 |\n| 7 | 17 | 834 |\n| 7 | 18 | 440 |\n| 7 | 19 | 834 |\n| 8 | 9 | 941 |\n| 8 | 11 | 941 |\n| 8 | 12 | 941 |\n| 8 | 13 | 941 |\n| 8 | 14 | 941 |\n| 8 | 15 | 861 |\n| 8 | 16 | 440 |\n| 8 | 17 | 323 |\n| 8 | 18 | 551 |\n| 8 | 19 | 1154 |\n| 9 | 10 | 671 |\n| 9 | 11 | 823 |\n| 9 | 13 | 1088 |\n| 9 | 14 | 182 |\n| 9 | 15 | 182 |\n| 9 | 16 | 182 |\n| 9 | 18 | 182 |\n| 9 | 19 | 182 |\n| 10 | 11 | 1295 |\n| 10 | 12 | 1295 |\n| 10 | 13 | 675 |\n| 10 | 14 | 675 |\n| 10 | 15 | 506 |\n| 10 | 18 | 675 |\n| 10 | 19 | 675 |\n| 11 | 12 | 482 |\n| 11 | 13 | 1286 |\n| 11 | 14 | 874 |\n| 11 | 15 | 1013 |\n| 11 | 18 | 313 |\n| 11 | 19 | 1286 |\n| 12 | 13 | 666 |\n| 12 | 16 | 430 |\n| 12 | 18 | 506 |\n| 12 | 19 | 506 |\n| 13 | 16 | 222 |\n| 13 | 17 | 640 |\n| 13 | 18 | 453 |\n| 13 | 19 | 482 |\n| 14 | 15 | 1252 |\n| 14 | 16 | 1571 |\n| 14 | 17 | 468 |\n| 14 | 18 | 468 |\n| 15 | 16 | 1433 |\n| 15 | 18 | 1113 |\n| 16 | 17 | 1113 |\n| 16 | 18 | 816 |\n| 16 | 19 | 690 |\n\nThe ops team must choose modules so total memory stays within 15677 while maximizing the system's total value.\n\nAlso, when you send your chosen set of features, please use this simple JSON layout so it's easy to read and check:\n\n{\n  \"solution\": [<selected_item_id>, <selected_item_id>, ...]\n}\n\nHere \"solution\" is just the list of feature IDs you want switched on — the exact labels from the instance input, in a plain array. This JSON is only a sketch of the shape I expect, not the actual answer itself.\n\nMake sure to use the identifiers exactly as they appear in the instance input — no renaming and no new labels.\n\nValid identifiers look like plain numbers such as \"1\" or \"23\", single capital letters like \"A\" or \"B\", or a capital letter followed by digits like \"A1\" or \"X7\".",
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    "instruction": "Someone in the school asked for a standout display, so the librarian is choosing which books to put on the front shelf to make the learning impact as strong as possible. Every chosen title uses up some of the shelf’s length and contributes its own educational points, and a few specific pairs of titles interact to give bonus points when both are present. A great arrangement keeps the total widths within the shelf limit and scores highly by totaling each book’s points plus any pair bonuses — with each book available as one copy only, and no partial placements allowed. The full details (widths, values, and which pairs get bonuses) are listed below.\n\n# total_books=21\n# shelf_length=6323\n# book_ids=1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21\n\nbook_id,educational_points\n1,132\n2,179\n3,102\n4,407\n5,230\n6,200\n7,470\n8,303\n9,250\n10,308\n11,264\n12,425\n13,116\n14,116\n15,422\n16,211\n17,183\n18,421\n19,406\n20,180\n21,435\n\nbook_id,width\n1,420\n2,343\n3,451\n4,304\n5,380\n6,170\n7,431\n8,321\n9,249\n10,380\n11,137\n12,103\n13,149\n14,353\n15,236\n16,360\n17,349\n18,350\n19,335\n20,313\n21,369\n\nbook_i_id,book_j_id,pairing_bonus_points\n1,2,318\n1,3,126\n1,4,468\n1,5,135\n1,6,380\n1,7,239\n1,8,494\n1,9,355\n1,10,217\n1,11,469\n1,12,219\n1,13,371\n1,15,232\n1,18,384\n1,19,170\n1,20,212\n1,21,264\n2,3,388\n2,4,129\n2,5,183\n2,6,255\n2,7,274\n2,8,271\n2,10,351\n2,11,307\n2,12,294\n2,13,100\n2,15,455\n2,16,326\n2,18,128\n2,20,418\n2,21,453\n3,4,177\n3,5,452\n3,6,494\n3,8,195\n3,9,396\n3,10,170\n3,11,254\n3,12,316\n3,14,228\n3,15,414\n3,18,235\n3,20,294\n4,5,478\n4,6,329\n4,7,163\n4,8,470\n4,11,422\n4,12,244\n4,13,350\n4,16,280\n4,17,101\n4,18,276\n4,19,241\n4,20,461\n4,21,414\n5,6,331\n5,7,375\n5,8,295\n5,9,369\n5,10,452\n5,11,248\n5,12,493\n5,13,208\n5,14,166\n5,15,223\n5,17,412\n5,18,260\n5,19,311\n5,20,122\n6,7,430\n6,8,139\n6,9,143\n6,11,274\n6,12,200\n6,13,447\n6,16,290\n6,18,453\n6,19,467\n6,21,233\n7,8,463\n7,9,291\n7,10,422\n7,11,185\n7,12,487\n7,13,258\n7,14,254\n7,15,166\n7,16,195\n7,17,374\n7,18,493\n7,19,269\n7,20,286\n7,21,456\n8,10,220\n8,11,309\n8,13,404\n8,14,433\n8,15,231\n8,17,185\n8,18,212\n8,19,178\n8,20,461\n8,21,282\n9,10,402\n9,13,324\n9,14,252\n9,17,438\n9,19,470\n10,11,295\n10,13,455\n10,14,214\n10,15,127\n10,16,262\n10,17,420\n10,18,299\n10,20,385\n10,21,433\n11,12,223\n11,13,317\n11,15,176\n11,16,362\n11,18,389\n11,19,120\n11,20,266\n11,21,454\n12,13,346\n12,14,155\n12,15,487\n12,16,376\n12,17,340\n12,18,484\n12,20,233\n12,21,252\n13,14,136\n13,15,145\n13,16,226\n13,17,328\n13,18,150\n13,19,418\n13,21,428\n14,15,298\n14,16,497\n14,18,112\n14,20,421\n14,21,173\n15,16,451\n15,17,315\n15,18,274\n15,19,166\n15,20,246\n16,17,340\n16,18,375\n16,19,127\n16,20,480\n16,21,246\n17,18,455\n17,19,242\n17,20,377\n17,21,231\n18,19,206\n18,20,488\n19,20,427\n19,21,484\n\nOh, and when you send back your pick for the front shelf, please use this simple JSON shape so it's easy to check automatically:\n\n{\n  \"solution\": [<book_id>, <book_id>, ...]\n}\n\nHere \"solution\" is just the list of the books you want on the front shelf — each entry is the identifier for a book from the instance. Think of it like filling out a short form: put the exact book IDs you chose inside the brackets, separated by commas. This is just a sketch of the shape we expect, not the final answer itself.\n\nPlease make sure to use the identifiers exactly as they appear in the instance input — no renaming and no new labels. Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.",
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  {
    "task_name": "QKP",
    "problem_type": "QKP",
    "instruction": "Many people on the scheduling desk worry about squeezing the best lineup into a fixed time slot. Every show uses up some minutes and earns a baseline audience, and there are a few beneficial pairings that add extra viewers when those two shows share the night. The job: choose a set of distinct shows that fits inside the block’s total time and maximizes the overall audience, which is just the sum of each show’s base viewers plus any applicable boosts from paired shows. The full runtimes, base ratings, and pairing bonuses appear below.\n\nThere are 20 shows available (IDs: A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T) and the evening block allows up to 1215 minutes.\nShow A draws a baseline audience of 123 viewers.\nShow B draws a baseline audience of 112 viewers.\nShow C draws a baseline audience of 101 viewers.\nShow D draws a baseline audience of 101 viewers.\nShow E draws a baseline audience of 122 viewers.\nShow F draws a baseline audience of 103 viewers.\nShow G draws a baseline audience of 110 viewers.\nShow H draws a baseline audience of 118 viewers.\nShow I draws a baseline audience of 106 viewers.\nShow J draws a baseline audience of 114 viewers.\nShow K draws a baseline audience of 102 viewers.\nShow L draws a baseline audience of 112 viewers.\nShow M draws a baseline audience of 111 viewers.\nShow N draws a baseline audience of 106 viewers.\nShow O draws a baseline audience of 123 viewers.\nShow P draws a baseline audience of 103 viewers.\nShow Q draws a baseline audience of 107 viewers.\nShow R draws a baseline audience of 110 viewers.\nShow S draws a baseline audience of 117 viewers.\nShow T draws a baseline audience of 102 viewers.\nShow A occupies 111 minutes of the block.\nShow B occupies 119 minutes of the block.\nShow C occupies 100 minutes of the block.\nShow D occupies 119 minutes of the block.\nShow E occupies 114 minutes of the block.\nShow F occupies 119 minutes of the block.\nShow G occupies 117 minutes of the block.\nShow H occupies 114 minutes of the block.\nShow I occupies 122 minutes of the block.\nShow J occupies 119 minutes of the block.\nShow K occupies 105 minutes of the block.\nShow L occupies 104 minutes of the block.\nShow M occupies 108 minutes of the block.\nShow N occupies 120 minutes of the block.\nShow O occupies 107 minutes of the block.\nShow P occupies 113 minutes of the block.\nShow Q occupies 107 minutes of the block.\nShow R occupies 116 minutes of the block.\nShow S occupies 104 minutes of the block.\nShow T occupies 120 minutes of the block.\nIf both A and B air, their pairing adds 120 viewers.\nIf both A and C air, their pairing adds 105 viewers.\nIf both A and D air, their pairing adds 119 viewers.\nIf both A and E air, their pairing adds 100 viewers.\nIf both A and F air, their pairing adds 122 viewers.\nIf both A and G air, their pairing adds 124 viewers.\nIf both A and H air, their pairing adds 102 viewers.\nIf both A and I air, their pairing adds 119 viewers.\nIf both A and J air, their pairing adds 123 viewers.\nIf both A and K air, their pairing adds 110 viewers.\nIf both A and L air, their pairing adds 102 viewers.\nIf both A and M air, their pairing adds 118 viewers.\nIf both A and N air, their pairing adds 114 viewers.\nIf both A and O air, their pairing adds 119 viewers.\nIf both A and P air, their pairing adds 105 viewers.\nIf both A and Q air, their pairing adds 111 viewers.\nIf both A and R air, their pairing adds 116 viewers.\nIf both A and S air, their pairing adds 112 viewers.\nIf both A and T air, their pairing adds 108 viewers.\nIf both B and C air, their pairing adds 101 viewers.\nIf both B and D air, their pairing adds 110 viewers.\nIf both B and E air, their pairing adds 116 viewers.\nIf both B and F air, their pairing adds 100 viewers.\nIf both B and G air, their pairing adds 112 viewers.\nIf both B and H air, their pairing adds 105 viewers.\nIf both B and I air, their pairing adds 104 viewers.\nIf both B and J air, their pairing adds 116 viewers.\nIf both B and K air, their pairing adds 111 viewers.\nIf both B and L air, their pairing adds 105 viewers.\nIf both B and M air, their pairing adds 102 viewers.\nIf both B and N air, their pairing adds 102 viewers.\nIf both B and O air, their pairing adds 104 viewers.\nIf both B and P air, their pairing adds 121 viewers.\nIf both B and Q air, their pairing adds 102 viewers.\nIf both B and R air, their pairing adds 117 viewers.\nIf both B and S air, their pairing adds 109 viewers.\nIf both B and T air, their pairing adds 108 viewers.\nIf both C and D air, their pairing adds 122 viewers.\nIf both C and E air, their pairing adds 104 viewers.\nIf both C and F air, their pairing adds 121 viewers.\nIf both C and G air, their pairing adds 119 viewers.\nIf both C and H air, their pairing adds 119 viewers.\nIf both C and I air, their pairing adds 107 viewers.\nIf both C and J air, their pairing adds 123 viewers.\nIf both C and K air, their pairing adds 119 viewers.\nIf both C and L air, their pairing adds 120 viewers.\nIf both C and M air, their pairing adds 107 viewers.\nIf both C and N air, their pairing adds 110 viewers.\nIf both C and O air, their pairing adds 116 viewers.\nIf both C and P air, their pairing adds 119 viewers.\nIf both C and Q air, their pairing adds 100 viewers.\nIf both C and R air, their pairing adds 124 viewers.\nIf both C and S air, their pairing adds 106 viewers.\nIf both C and T air, their pairing adds 100 viewers.\nIf both D and E air, their pairing adds 124 viewers.\nIf both D and F air, their pairing adds 111 viewers.\nIf both D and G air, their pairing adds 124 viewers.\nIf both D and H air, their pairing adds 112 viewers.\nIf both D and I air, their pairing adds 110 viewers.\nIf both D and J air, their pairing adds 116 viewers.\nIf both D and K air, their pairing adds 119 viewers.\nIf both D and L air, their pairing adds 112 viewers.\nIf both D and M air, their pairing adds 109 viewers.\nIf both D and N air, their pairing adds 100 viewers.\nIf both D and O air, their pairing adds 102 viewers.\nIf both D and P air, their pairing adds 118 viewers.\nIf both D and Q air, their pairing adds 103 viewers.\nIf both D and R air, their pairing adds 112 viewers.\nIf both D and S air, their pairing adds 118 viewers.\nIf both D and T air, their pairing adds 118 viewers.\nIf both E and F air, their pairing adds 101 viewers.\nIf both E and G air, their pairing adds 123 viewers.\nIf both E and H air, their pairing adds 110 viewers.\nIf both E and I air, their pairing adds 103 viewers.\nIf both E and J air, their pairing adds 102 viewers.\nIf both E and K air, their pairing adds 121 viewers.\nIf both E and L air, their pairing adds 102 viewers.\nIf both E and M air, their pairing adds 102 viewers.\nIf both E and N air, their pairing adds 105 viewers.\nIf both E and O air, their pairing adds 100 viewers.\nIf both E and P air, their pairing adds 120 viewers.\nIf both E and Q air, their pairing adds 121 viewers.\nIf both E and R air, their pairing adds 118 viewers.\nIf both E and S air, their pairing adds 117 viewers.\nIf both E and T air, their pairing adds 109 viewers.\nIf both F and G air, their pairing adds 119 viewers.\nIf both F and H air, their pairing adds 121 viewers.\nIf both F and I air, their pairing adds 118 viewers.\nIf both F and J air, their pairing adds 101 viewers.\nIf both F and K air, their pairing adds 104 viewers.\nIf both F and L air, their pairing adds 112 viewers.\nIf both F and M air, their pairing adds 106 viewers.\nIf both F and N air, their pairing adds 109 viewers.\nIf both F and O air, their pairing adds 124 viewers.\nIf both F and P air, their pairing adds 102 viewers.\nIf both F and Q air, their pairing adds 114 viewers.\nIf both F and R air, their pairing adds 112 viewers.\nIf both F and S air, their pairing adds 112 viewers.\nIf both F and T air, their pairing adds 110 viewers.\nIf both G and H air, their pairing adds 113 viewers.\nIf both G and I air, their pairing adds 124 viewers.\nIf both G and J air, their pairing adds 106 viewers.\nIf both G and K air, their pairing adds 111 viewers.\nIf both G and L air, their pairing adds 107 viewers.\nIf both G and M air, their pairing adds 111 viewers.\nIf both G and N air, their pairing adds 108 viewers.\nIf both G and O air, their pairing adds 118 viewers.\nIf both G and P air, their pairing adds 110 viewers.\nIf both G and Q air, their pairing adds 107 viewers.\nIf both G and R air, their pairing adds 124 viewers.\nIf both G and S air, their pairing adds 121 viewers.\nIf both G and T air, their pairing adds 103 viewers.\nIf both H and I air, their pairing adds 104 viewers.\nIf both H and J air, their pairing adds 111 viewers.\nIf both H and K air, their pairing adds 113 viewers.\nIf both H and L air, their pairing adds 107 viewers.\nIf both H and M air, their pairing adds 115 viewers.\nIf both H and N air, their pairing adds 121 viewers.\nIf both H and O air, their pairing adds 101 viewers.\nIf both H and P air, their pairing adds 102 viewers.\nIf both H and Q air, their pairing adds 122 viewers.\nIf both H and R air, their pairing adds 123 viewers.\nIf both H and S air, their pairing adds 111 viewers.\nIf both H and T air, their pairing adds 124 viewers.\nIf both I and J air, their pairing adds 115 viewers.\nIf both I and K air, their pairing adds 104 viewers.\nIf both I and L air, their pairing adds 121 viewers.\nIf both I and M air, their pairing adds 124 viewers.\nIf both I and N air, their pairing adds 114 viewers.\nIf both I and O air, their pairing adds 108 viewers.\nIf both I and P air, their pairing adds 107 viewers.\nIf both I and Q air, their pairing adds 107 viewers.\nIf both I and R air, their pairing adds 114 viewers.\nIf both I and S air, their pairing adds 109 viewers.\nIf both I and T air, their pairing adds 118 viewers.\nIf both J and K air, their pairing adds 103 viewers.\nIf both J and L air, their pairing adds 103 viewers.\nIf both J and M air, their pairing adds 121 viewers.\nIf both J and N air, their pairing adds 108 viewers.\nIf both J and O air, their pairing adds 113 viewers.\nIf both J and P air, their pairing adds 114 viewers.\nIf both J and Q air, their pairing adds 109 viewers.\nIf both J and R air, their pairing adds 115 viewers.\nIf both J and S air, their pairing adds 103 viewers.\nIf both J and T air, their pairing adds 105 viewers.\nIf both K and L air, their pairing adds 109 viewers.\nIf both K and M air, their pairing adds 118 viewers.\nIf both K and N air, their pairing adds 114 viewers.\nIf both K and O air, their pairing adds 110 viewers.\nIf both K and P air, their pairing adds 111 viewers.\nIf both K and Q air, their pairing adds 101 viewers.\nIf both K and R air, their pairing adds 108 viewers.\nIf both K and S air, their pairing adds 123 viewers.\nIf both K and T air, their pairing adds 105 viewers.\nIf both L and M air, their pairing adds 101 viewers.\nIf both L and N air, their pairing adds 100 viewers.\nIf both L and O air, their pairing adds 106 viewers.\nIf both L and P air, their pairing adds 107 viewers.\nIf both L and Q air, their pairing adds 100 viewers.\nIf both L and R air, their pairing adds 121 viewers.\nIf both L and S air, their pairing adds 105 viewers.\nIf both L and T air, their pairing adds 122 viewers.\nIf both M and N air, their pairing adds 123 viewers.\nIf both M and O air, their pairing adds 121 viewers.\nIf both M and P air, their pairing adds 105 viewers.\nIf both M and Q air, their pairing adds 106 viewers.\nIf both M and R air, their pairing adds 124 viewers.\nIf both M and S air, their pairing adds 107 viewers.\nIf both M and T air, their pairing adds 104 viewers.\nIf both N and O air, their pairing adds 115 viewers.\nIf both N and P air, their pairing adds 115 viewers.\nIf both N and Q air, their pairing adds 119 viewers.\nIf both N and R air, their pairing adds 124 viewers.\nIf both N and S air, their pairing adds 111 viewers.\nIf both N and T air, their pairing adds 111 viewers.\nIf both O and P air, their pairing adds 113 viewers.\nIf both O and Q air, their pairing adds 100 viewers.\nIf both O and R air, their pairing adds 104 viewers.\nIf both O and S air, their pairing adds 105 viewers.\nIf both O and T air, their pairing adds 120 viewers.\nIf both P and Q air, their pairing adds 104 viewers.\nIf both P and R air, their pairing adds 101 viewers.\nIf both P and S air, their pairing adds 109 viewers.\nIf both P and T air, their pairing adds 103 viewers.\nIf both Q and R air, their pairing adds 123 viewers.\nIf both Q and S air, their pairing adds 106 viewers.\nIf both Q and T air, their pairing adds 109 viewers.\nIf both R and S air, their pairing adds 105 viewers.\nIf both R and T air, their pairing adds 100 viewers.\nIf both S and T air, their pairing adds 100 viewers.\nChoose a lineup that fits within 1215 minutes to maximize total viewers from base audiences and any pairing uplifts.\n\nOh, and when you send back your pick, just stick to a tiny JSON sketch like this so it's easy to read and check:\n\n{\n  \"solution\": [<selected_show_id>, <selected_show_id>, ...]\n}\n\nThis just means \"solution\" should be a list of the shows you choose (use each show's identifier from the instance). It's just a shape guide — not the actual lineup — so fill in the real show IDs when you reply.\n\nPlease use the exact identifiers from the instance input, with no renaming or invented labels.\n- for example: \"Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.\"",
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          "item_i_id": "L",
          "item_j_id": "L",
          "quadratic_profit": 123
        },
        {
          "item_i_id": "L",
          "item_j_id": "M",
          "quadratic_profit": 101
        },
        {
          "item_i_id": "L",
          "item_j_id": "N",
          "quadratic_profit": 100
        },
        {
          "item_i_id": "L",
          "item_j_id": "O",
          "quadratic_profit": 106
        },
        {
          "item_i_id": "L",
          "item_j_id": "P",
          "quadratic_profit": 107
        },
        {
          "item_i_id": "L",
          "item_j_id": "Q",
          "quadratic_profit": 100
        },
        {
          "item_i_id": "L",
          "item_j_id": "R",
          "quadratic_profit": 121
        },
        {
          "item_i_id": "L",
          "item_j_id": "S",
          "quadratic_profit": 105
        },
        {
          "item_i_id": "L",
          "item_j_id": "T",
          "quadratic_profit": 122
        },
        {
          "item_i_id": "M",
          "item_j_id": "M",
          "quadratic_profit": 114
        },
        {
          "item_i_id": "M",
          "item_j_id": "N",
          "quadratic_profit": 123
        },
        {
          "item_i_id": "M",
          "item_j_id": "O",
          "quadratic_profit": 121
        },
        {
          "item_i_id": "M",
          "item_j_id": "P",
          "quadratic_profit": 105
        },
        {
          "item_i_id": "M",
          "item_j_id": "Q",
          "quadratic_profit": 106
        },
        {
          "item_i_id": "M",
          "item_j_id": "R",
          "quadratic_profit": 124
        },
        {
          "item_i_id": "M",
          "item_j_id": "S",
          "quadratic_profit": 107
        },
        {
          "item_i_id": "M",
          "item_j_id": "T",
          "quadratic_profit": 104
        },
        {
          "item_i_id": "N",
          "item_j_id": "N",
          "quadratic_profit": 122
        },
        {
          "item_i_id": "N",
          "item_j_id": "O",
          "quadratic_profit": 115
        },
        {
          "item_i_id": "N",
          "item_j_id": "P",
          "quadratic_profit": 115
        },
        {
          "item_i_id": "N",
          "item_j_id": "Q",
          "quadratic_profit": 119
        },
        {
          "item_i_id": "N",
          "item_j_id": "R",
          "quadratic_profit": 124
        },
        {
          "item_i_id": "N",
          "item_j_id": "S",
          "quadratic_profit": 111
        },
        {
          "item_i_id": "N",
          "item_j_id": "T",
          "quadratic_profit": 111
        },
        {
          "item_i_id": "O",
          "item_j_id": "O",
          "quadratic_profit": 120
        },
        {
          "item_i_id": "O",
          "item_j_id": "P",
          "quadratic_profit": 113
        },
        {
          "item_i_id": "O",
          "item_j_id": "Q",
          "quadratic_profit": 100
        },
        {
          "item_i_id": "O",
          "item_j_id": "R",
          "quadratic_profit": 104
        },
        {
          "item_i_id": "O",
          "item_j_id": "S",
          "quadratic_profit": 105
        },
        {
          "item_i_id": "O",
          "item_j_id": "T",
          "quadratic_profit": 120
        },
        {
          "item_i_id": "P",
          "item_j_id": "P",
          "quadratic_profit": 101
        },
        {
          "item_i_id": "P",
          "item_j_id": "Q",
          "quadratic_profit": 104
        },
        {
          "item_i_id": "P",
          "item_j_id": "R",
          "quadratic_profit": 101
        },
        {
          "item_i_id": "P",
          "item_j_id": "S",
          "quadratic_profit": 109
        },
        {
          "item_i_id": "P",
          "item_j_id": "T",
          "quadratic_profit": 103
        },
        {
          "item_i_id": "Q",
          "item_j_id": "Q",
          "quadratic_profit": 113
        },
        {
          "item_i_id": "Q",
          "item_j_id": "R",
          "quadratic_profit": 123
        },
        {
          "item_i_id": "Q",
          "item_j_id": "S",
          "quadratic_profit": 106
        },
        {
          "item_i_id": "Q",
          "item_j_id": "T",
          "quadratic_profit": 109
        },
        {
          "item_i_id": "R",
          "item_j_id": "R",
          "quadratic_profit": 102
        },
        {
          "item_i_id": "R",
          "item_j_id": "S",
          "quadratic_profit": 105
        },
        {
          "item_i_id": "R",
          "item_j_id": "T",
          "quadratic_profit": 100
        },
        {
          "item_i_id": "S",
          "item_j_id": "S",
          "quadratic_profit": 115
        },
        {
          "item_i_id": "S",
          "item_j_id": "T",
          "quadratic_profit": 100
        }
      ]
    },
    "solution_variant": [
      "A",
      "C",
      "F",
      "G",
      "K",
      "M",
      "N",
      "O",
      "Q",
      "R",
      "S"
    ],
    "context_index": 30,
    "input_format": "nl",
    "input_index_base": "names"
  },
  {
    "task_name": "QKP",
    "problem_type": "QKP",
    "instruction": "Out on the river the difference between a good day and a great day can be what’s in the backpack. Every rod, lure and tool adds weight and gives a baseline chance of scoring fish, and a few specific pairings boost that chance when they travel together. The plan is to choose a combination of gear whose combined weight fits within the pack’s limit and whose total expected catch value equals the sum of all chosen items’ base values plus any pairwise bonuses, with no item repeated. The detailed item list and weight/bonus numbers are below.\n\nBelow are 21 tackle options listed as 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, and the pack's limit is 1255.\nThe tackle 1 contributes a baseline expected catch value of 103.\nThe tackle 2 contributes a baseline expected catch value of 100.\nThe tackle 3 contributes a baseline expected catch value of 104.\nThe tackle 4 contributes a baseline expected catch value of 104.\nThe tackle 5 contributes a baseline expected catch value of 104.\nThe tackle 6 contributes a baseline expected catch value of 105.\nThe tackle 7 contributes a baseline expected catch value of 103.\nThe tackle 8 contributes a baseline expected catch value of 105.\nThe tackle 9 contributes a baseline expected catch value of 105.\nThe tackle 10 contributes a baseline expected catch value of 100.\nThe tackle 11 contributes a baseline expected catch value of 102.\nThe tackle 12 contributes a baseline expected catch value of 102.\nThe tackle 13 contributes a baseline expected catch value of 105.\nThe tackle 14 contributes a baseline expected catch value of 103.\nThe tackle 15 contributes a baseline expected catch value of 102.\nThe tackle 16 contributes a baseline expected catch value of 102.\nThe tackle 17 contributes a baseline expected catch value of 100.\nThe tackle 18 contributes a baseline expected catch value of 105.\nThe tackle 19 contributes a baseline expected catch value of 104.\nThe tackle 20 contributes a baseline expected catch value of 103.\nThe tackle 21 contributes a baseline expected catch value of 104.\n1 weighs 101, adding to the pack's total load.\n2 weighs 103, adding to the pack's total load.\n3 weighs 103, adding to the pack's total load.\n4 weighs 102, adding to the pack's total load.\n5 weighs 105, adding to the pack's total load.\n6 weighs 103, adding to the pack's total load.\n7 weighs 105, adding to the pack's total load.\n8 weighs 103, adding to the pack's total load.\n9 weighs 105, adding to the pack's total load.\n10 weighs 102, adding to the pack's total load.\n11 weighs 103, adding to the pack's total load.\n12 weighs 102, adding to the pack's total load.\n13 weighs 100, adding to the pack's total load.\n14 weighs 105, adding to the pack's total load.\n15 weighs 101, adding to the pack's total load.\n16 weighs 102, adding to the pack's total load.\n17 weighs 100, adding to the pack's total load.\n18 weighs 103, adding to the pack's total load.\n19 weighs 100, adding to the pack's total load.\n20 weighs 100, adding to the pack's total load.\n21 weighs 100, adding to the pack's total load.\nIncluding 1 together with 2 yields a combination bonus of 101.\nIncluding 1 together with 3 yields a combination bonus of 102.\nIncluding 1 together with 4 yields a combination bonus of 101.\nIncluding 1 together with 5 yields a combination bonus of 101.\nIncluding 1 together with 7 yields a combination bonus of 102.\nIncluding 1 together with 8 yields a combination bonus of 102.\nIncluding 1 together with 11 yields a combination bonus of 102.\nIncluding 1 together with 13 yields a combination bonus of 102.\nIncluding 1 together with 15 yields a combination bonus of 101.\nIncluding 1 together with 17 yields a combination bonus of 100.\nIncluding 1 together with 18 yields a combination bonus of 100.\nIncluding 1 together with 19 yields a combination bonus of 100.\nIncluding 1 together with 20 yields a combination bonus of 102.\nIncluding 1 together with 21 yields a combination bonus of 102.\nIncluding 2 together with 3 yields a combination bonus of 101.\nIncluding 2 together with 4 yields a combination bonus of 103.\nIncluding 2 together with 5 yields a combination bonus of 102.\nIncluding 2 together with 7 yields a combination bonus of 103.\nIncluding 2 together with 8 yields a combination bonus of 101.\nIncluding 2 together with 10 yields a combination bonus of 101.\nIncluding 2 together with 11 yields a combination bonus of 102.\nIncluding 2 together with 12 yields a combination bonus of 101.\nIncluding 2 together with 15 yields a combination bonus of 103.\nIncluding 2 together with 16 yields a combination bonus of 103.\nIncluding 2 together with 17 yields a combination bonus of 101.\nIncluding 2 together with 18 yields a combination bonus of 102.\nIncluding 2 together with 20 yields a combination bonus of 102.\nIncluding 2 together with 21 yields a combination bonus of 103.\nIncluding 3 together with 4 yields a combination bonus of 103.\nIncluding 3 together with 5 yields a combination bonus of 102.\nIncluding 3 together with 7 yields a combination bonus of 101.\nIncluding 3 together with 9 yields a combination bonus of 102.\nIncluding 3 together with 10 yields a combination bonus of 103.\nIncluding 3 together with 11 yields a combination bonus of 101.\nIncluding 3 together with 12 yields a combination bonus of 103.\nIncluding 3 together with 14 yields a combination bonus of 102.\nIncluding 3 together with 15 yields a combination bonus of 103.\nIncluding 3 together with 16 yields a combination bonus of 101.\nIncluding 3 together with 17 yields a combination bonus of 102.\nIncluding 3 together with 18 yields a combination bonus of 102.\nIncluding 3 together with 19 yields a combination bonus of 101.\nIncluding 3 together with 20 yields a combination bonus of 102.\nIncluding 4 together with 6 yields a combination bonus of 102.\nIncluding 4 together with 7 yields a combination bonus of 103.\nIncluding 4 together with 8 yields a combination bonus of 102.\nIncluding 4 together with 9 yields a combination bonus of 100.\nIncluding 4 together with 12 yields a combination bonus of 102.\nIncluding 4 together with 13 yields a combination bonus of 102.\nIncluding 4 together with 15 yields a combination bonus of 103.\nIncluding 4 together with 16 yields a combination bonus of 100.\nIncluding 4 together with 18 yields a combination bonus of 101.\nIncluding 4 together with 19 yields a combination bonus of 102.\nIncluding 4 together with 20 yields a combination bonus of 103.\nIncluding 4 together with 21 yields a combination bonus of 102.\nIncluding 5 together with 7 yields a combination bonus of 104.\nIncluding 5 together with 8 yields a combination bonus of 103.\nIncluding 5 together with 9 yields a combination bonus of 104.\nIncluding 5 together with 10 yields a combination bonus of 104.\nIncluding 5 together with 11 yields a combination bonus of 105.\nIncluding 5 together with 12 yields a combination bonus of 102.\nIncluding 5 together with 13 yields a combination bonus of 102.\nIncluding 5 together with 14 yields a combination bonus of 103.\nIncluding 5 together with 15 yields a combination bonus of 102.\nIncluding 5 together with 16 yields a combination bonus of 101.\nIncluding 5 together with 17 yields a combination bonus of 101.\nIncluding 5 together with 18 yields a combination bonus of 100.\nIncluding 5 together with 19 yields a combination bonus of 101.\nIncluding 5 together with 20 yields a combination bonus of 101.\nIncluding 5 together with 21 yields a combination bonus of 101.\nIncluding 6 together with 7 yields a combination bonus of 103.\nIncluding 6 together with 8 yields a combination bonus of 101.\nIncluding 6 together with 10 yields a combination bonus of 101.\nIncluding 6 together with 11 yields a combination bonus of 103.\nIncluding 6 together with 12 yields a combination bonus of 102.\nIncluding 6 together with 13 yields a combination bonus of 103.\nIncluding 6 together with 14 yields a combination bonus of 103.\nIncluding 6 together with 15 yields a combination bonus of 101.\nIncluding 6 together with 16 yields a combination bonus of 101.\nIncluding 6 together with 17 yields a combination bonus of 103.\nIncluding 6 together with 18 yields a combination bonus of 102.\nIncluding 6 together with 19 yields a combination bonus of 103.\nIncluding 7 together with 8 yields a combination bonus of 105.\nIncluding 7 together with 9 yields a combination bonus of 102.\nIncluding 7 together with 10 yields a combination bonus of 105.\nIncluding 7 together with 11 yields a combination bonus of 104.\nIncluding 7 together with 13 yields a combination bonus of 102.\nIncluding 7 together with 14 yields a combination bonus of 102.\nIncluding 7 together with 15 yields a combination bonus of 103.\nIncluding 7 together with 16 yields a combination bonus of 101.\nIncluding 7 together with 17 yields a combination bonus of 102.\nIncluding 7 together with 18 yields a combination bonus of 102.\nIncluding 7 together with 20 yields a combination bonus of 102.\nIncluding 8 together with 9 yields a combination bonus of 101.\nIncluding 8 together with 10 yields a combination bonus of 103.\nIncluding 8 together with 12 yields a combination bonus of 101.\nIncluding 8 together with 13 yields a combination bonus of 101.\nIncluding 8 together with 14 yields a combination bonus of 101.\nIncluding 8 together with 15 yields a combination bonus of 103.\nIncluding 8 together with 16 yields a combination bonus of 104.\nIncluding 8 together with 17 yields a combination bonus of 102.\nIncluding 8 together with 21 yields a combination bonus of 104.\nIncluding 9 together with 12 yields a combination bonus of 102.\nIncluding 9 together with 13 yields a combination bonus of 101.\nIncluding 9 together with 15 yields a combination bonus of 100.\nIncluding 9 together with 16 yields a combination bonus of 100.\nIncluding 9 together with 17 yields a combination bonus of 100.\nIncluding 9 together with 19 yields a combination bonus of 102.\nIncluding 9 together with 20 yields a combination bonus of 100.\nIncluding 9 together with 21 yields a combination bonus of 101.\nIncluding 10 together with 11 yields a combination bonus of 100.\nIncluding 10 together with 12 yields a combination bonus of 101.\nIncluding 10 together with 13 yields a combination bonus of 102.\nIncluding 10 together with 14 yields a combination bonus of 102.\nIncluding 10 together with 15 yields a combination bonus of 104.\nIncluding 10 together with 18 yields a combination bonus of 102.\nIncluding 10 together with 20 yields a combination bonus of 103.\nIncluding 11 together with 12 yields a combination bonus of 101.\nIncluding 11 together with 13 yields a combination bonus of 101.\nIncluding 11 together with 14 yields a combination bonus of 102.\nIncluding 11 together with 15 yields a combination bonus of 100.\nIncluding 11 together with 17 yields a combination bonus of 100.\nIncluding 11 together with 18 yields a combination bonus of 100.\nIncluding 11 together with 19 yields a combination bonus of 100.\nIncluding 11 together with 20 yields a combination bonus of 102.\nIncluding 12 together with 15 yields a combination bonus of 100.\nIncluding 12 together with 16 yields a combination bonus of 100.\nIncluding 12 together with 17 yields a combination bonus of 101.\nIncluding 12 together with 19 yields a combination bonus of 102.\nIncluding 12 together with 21 yields a combination bonus of 100.\nIncluding 13 together with 14 yields a combination bonus of 103.\nIncluding 13 together with 15 yields a combination bonus of 103.\nIncluding 13 together with 16 yields a combination bonus of 102.\nIncluding 13 together with 19 yields a combination bonus of 103.\nIncluding 13 together with 20 yields a combination bonus of 105.\nIncluding 13 together with 21 yields a combination bonus of 105.\nIncluding 14 together with 15 yields a combination bonus of 101.\nIncluding 14 together with 17 yields a combination bonus of 101.\nIncluding 14 together with 18 yields a combination bonus of 100.\nIncluding 14 together with 19 yields a combination bonus of 101.\nIncluding 15 together with 19 yields a combination bonus of 101.\nIncluding 15 together with 20 yields a combination bonus of 101.\nIncluding 15 together with 21 yields a combination bonus of 103.\nPack smart: choose gear and pairings that raise total expected catch while staying within the 1255 limit.\n\nOh, and when you’re ready to tell me which gear you’d pack, just drop it in this simple JSON shape so I can read it cleanly:\n\n{\n  \"solution\": [<selected_item_id>, <selected_item_id>, ...]\n}\n\nThe \"solution\" array is just the list of the item IDs you want to take in your pack. Think of it like ticking off the gear names on a checklist — nothing fancy, just the exact labels for the items you chose. This is only a sketch of the expected shape, not the actual answer itself.\n\nPlease use the item identifiers exactly as they appear in the instance input — no renaming and no inventing new labels. For example: \"Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.\"",
    "instance": {
      "linear_coeffs": [
        103,
        100,
        104,
        104,
        104,
        105,
        103,
        105,
        105,
        100,
        102,
        102,
        105,
        103,
        102,
        102,
        100,
        105,
        104,
        103,
        104
      ],
      "quadratic_coeffs": [
        [
          101,
          101,
          102,
          101,
          101,
          0,
          102,
          102,
          0,
          0,
          102,
          0,
          102,
          0,
          101,
          0,
          100,
          100,
          100,
          102,
          102
        ],
        [
          0,
          102,
          101,
          103,
          102,
          0,
          103,
          101,
          0,
          101,
          102,
          101,
          0,
          0,
          103,
          103,
          101,
          102,
          0,
          102,
          103
        ],
        [
          0,
          0,
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          103,
          102,
          0,
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          0,
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          103,
          101,
          103,
          0,
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          103,
          101,
          102,
          102,
          101,
          102,
          0
        ],
        [
          0,
          0,
          0,
          0,
          0,
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          102,
          100,
          0,
          0,
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        ],
        [
          0,
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          0,
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        ],
        [
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          0,
          0,
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          0,
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        ],
        [
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          0,
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        ],
        [
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        ],
        [
          0,
          0,
          0,
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          0,
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        ],
        [
          0,
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        ],
        [
          0,
          0,
          0,
          0,
          0,
          0,
          0,
          0,
          0,
          0,
          103,
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          0
        ],
        [
          0,
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          0,
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        ],
        [
          0,
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        ],
        [
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        ],
        [
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        ],
        [
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        ]
      ],
      "weights": [
        101,
        103,
        103,
        102,
        105,
        103,
        105,
        103,
        105,
        102,
        103,
        102,
        100,
        105,
        101,
        102,
        100,
        103,
        100,
        100,
        100
      ],
      "capacity": 1255,
      "solution": [
        0,
        1,
        2,
        3,
        4,
        6,
        7,
        9,
        10,
        12,
        14,
        19
      ],
      "obj": 7067.0,
      "problem_type": "QKP"
    },
    "solution": [
      0,
      1,
      2,
      3,
      4,
      6,
      7,
      9,
      10,
      12,
      14,
      19
    ],
    "obj": 7067.0,
    "instance_variant": {
      "problem_type": "QKP",
      "num_items": 21,
      "capacity": 1255,
      "items": [
        1,
        2,
        3,
        4,
        5,
        6,
        7,
        8,
        9,
        10,
        11,
        12,
        13,
        14,
        15,
        16,
        17,
        18,
        19,
        20,
        21
      ],
      "linear_pairs": [
        {
          "item_id": 1,
          "linear_profit": 103
        },
        {
          "item_id": 2,
          "linear_profit": 100
        },
        {
          "item_id": 3,
          "linear_profit": 104
        },
        {
          "item_id": 4,
          "linear_profit": 104
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    "context_index": 31,
    "input_format": "nl",
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  },
  {
    "task_name": "QKP",
    "problem_type": "QKP",
    "instruction": "We’re getting a travel-friendly portrait kit ready for a wedding day: every flash or modifier adds bulk but also contributes to the kinds of images that can be made, and some combos give bonus creative options when used together. The job is to choose which pieces to bring so the total weight fits the bag’s limit while the total benefit is the sum of each item’s standalone value plus any extra value from combos; each item can only be taken once. The concrete specs — weights, individual scores, and combo boosts — are listed below.\n\nWe have 16 pieces (0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15) and a bag limit of 876; choose which to pack so total weight fits the limit.\nTaking 0 adds 104 standalone value to our kit.\nTaking 1 adds 103 standalone value to our kit.\nTaking 2 adds 105 standalone value to our kit.\nTaking 3 adds 103 standalone value to our kit.\nTaking 4 adds 100 standalone value to our kit.\nTaking 5 adds 105 standalone value to our kit.\nTaking 6 adds 102 standalone value to our kit.\nTaking 7 adds 103 standalone value to our kit.\nTaking 8 adds 103 standalone value to our kit.\nTaking 9 adds 104 standalone value to our kit.\nTaking 10 adds 103 standalone value to our kit.\nTaking 11 adds 102 standalone value to our kit.\nTaking 12 adds 102 standalone value to our kit.\nTaking 13 adds 102 standalone value to our kit.\nTaking 14 adds 103 standalone value to our kit.\nTaking 15 adds 101 standalone value to our kit.\nTaking 0 adds 101 weight to our load.\nTaking 1 adds 100 weight to our load.\nTaking 2 adds 102 weight to our load.\nTaking 3 adds 101 weight to our load.\nTaking 4 adds 102 weight to our load.\nTaking 5 adds 105 weight to our load.\nTaking 6 adds 103 weight to our load.\nTaking 7 adds 104 weight to our load.\nTaking 8 adds 100 weight to our load.\nTaking 9 adds 102 weight to our load.\nTaking 10 adds 105 weight to our load.\nTaking 11 adds 100 weight to our load.\nTaking 12 adds 104 weight to our load.\nTaking 13 adds 102 weight to our load.\nTaking 14 adds 102 weight to our load.\nTaking 15 adds 101 weight to our load.\nBringing 0 with 1 yields an extra 101 creative bonus when both are packed.\nBringing 0 with 2 yields an extra 101 creative bonus when both are packed.\nBringing 0 with 3 yields an extra 101 creative bonus when both are packed.\nBringing 0 with 4 yields an extra 100 creative bonus when both are packed.\nBringing 0 with 5 yields an extra 100 creative bonus when both are packed.\nBringing 0 with 6 yields an extra 100 creative bonus when both are packed.\nBringing 0 with 7 yields an extra 100 creative bonus when both are packed.\nBringing 0 with 8 yields an extra 101 creative bonus when both are packed.\nBringing 0 with 9 yields an extra 101 creative bonus when both are packed.\nBringing 0 with 10 yields an extra 102 creative bonus when both are packed.\nBringing 0 with 11 yields an extra 101 creative bonus when both are packed.\nBringing 0 with 12 yields an extra 102 creative bonus when both are packed.\nBringing 0 with 13 yields an extra 101 creative bonus when both are packed.\nBringing 0 with 14 yields an extra 104 creative bonus when both are packed.\nBringing 0 with 15 yields an extra 102 creative bonus when both are packed.\nBringing 1 with 2 yields an extra 100 creative bonus when both are packed.\nBringing 1 with 3 yields an extra 100 creative bonus when both are packed.\nBringing 1 with 4 yields an extra 101 creative bonus when both are packed.\nBringing 1 with 5 yields an extra 100 creative bonus when both are packed.\nBringing 1 with 6 yields an extra 100 creative bonus when both are packed.\nBringing 1 with 7 yields an extra 100 creative bonus when both are packed.\nBringing 1 with 8 yields an extra 100 creative bonus when both are packed.\nBringing 1 with 9 yields an extra 100 creative bonus when both are packed.\nBringing 1 with 10 yields an extra 103 creative bonus when both are packed.\nBringing 1 with 11 yields an extra 103 creative bonus when both are packed.\nBringing 1 with 12 yields an extra 101 creative bonus when both are packed.\nBringing 1 with 13 yields an extra 103 creative bonus when both are packed.\nBringing 1 with 14 yields an extra 103 creative bonus when both are packed.\nBringing 1 with 15 yields an extra 103 creative bonus when both are packed.\nBringing 2 with 3 yields an extra 102 creative bonus when both are packed.\nBringing 2 with 4 yields an extra 102 creative bonus when both are packed.\nBringing 2 with 5 yields an extra 100 creative bonus when both are packed.\nBringing 2 with 6 yields an extra 102 creative bonus when both are packed.\nBringing 2 with 7 yields an extra 102 creative bonus when both are packed.\nBringing 2 with 8 yields an extra 102 creative bonus when both are packed.\nBringing 2 with 9 yields an extra 103 creative bonus when both are packed.\nBringing 2 with 10 yields an extra 101 creative bonus when both are packed.\nBringing 2 with 11 yields an extra 102 creative bonus when both are packed.\nBringing 2 with 12 yields an extra 102 creative bonus when both are packed.\nBringing 2 with 13 yields an extra 100 creative bonus when both are packed.\nBringing 2 with 14 yields an extra 100 creative bonus when both are packed.\nBringing 2 with 15 yields an extra 102 creative bonus when both are packed.\nBringing 3 with 4 yields an extra 101 creative bonus when both are packed.\nBringing 3 with 5 yields an extra 100 creative bonus when both are packed.\nBringing 3 with 6 yields an extra 102 creative bonus when both are packed.\nBringing 3 with 7 yields an extra 102 creative bonus when both are packed.\nBringing 3 with 8 yields an extra 101 creative bonus when both are packed.\nBringing 3 with 9 yields an extra 100 creative bonus when both are packed.\nBringing 3 with 10 yields an extra 100 creative bonus when both are packed.\nBringing 3 with 11 yields an extra 100 creative bonus when both are packed.\nBringing 3 with 12 yields an extra 101 creative bonus when both are packed.\nBringing 3 with 13 yields an extra 100 creative bonus when both are packed.\nBringing 3 with 14 yields an extra 100 creative bonus when both are packed.\nBringing 3 with 15 yields an extra 101 creative bonus when both are packed.\nBringing 4 with 5 yields an extra 102 creative bonus when both are packed.\nBringing 4 with 6 yields an extra 100 creative bonus when both are packed.\nBringing 4 with 7 yields an extra 102 creative bonus when both are packed.\nBringing 4 with 8 yields an extra 100 creative bonus when both are packed.\nBringing 4 with 9 yields an extra 100 creative bonus when both are packed.\nBringing 4 with 10 yields an extra 101 creative bonus when both are packed.\nBringing 4 with 11 yields an extra 101 creative bonus when both are packed.\nBringing 4 with 12 yields an extra 101 creative bonus when both are packed.\nBringing 4 with 13 yields an extra 100 creative bonus when both are packed.\nBringing 4 with 14 yields an extra 101 creative bonus when both are packed.\nBringing 4 with 15 yields an extra 101 creative bonus when both are packed.\nBringing 5 with 6 yields an extra 104 creative bonus when both are packed.\nBringing 5 with 7 yields an extra 104 creative bonus when both are packed.\nBringing 5 with 8 yields an extra 100 creative bonus when both are packed.\nBringing 5 with 9 yields an extra 100 creative bonus when both are packed.\nBringing 5 with 10 yields an extra 100 creative bonus when both are packed.\nBringing 5 with 11 yields an extra 100 creative bonus when both are packed.\nBringing 5 with 12 yields an extra 100 creative bonus when both are packed.\nBringing 5 with 13 yields an extra 100 creative bonus when both are packed.\nBringing 5 with 14 yields an extra 100 creative bonus when both are packed.\nBringing 5 with 15 yields an extra 100 creative bonus when both are packed.\nBringing 6 with 7 yields an extra 103 creative bonus when both are packed.\nBringing 6 with 8 yields an extra 104 creative bonus when both are packed.\nBringing 6 with 9 yields an extra 102 creative bonus when both are packed.\nBringing 6 with 10 yields an extra 102 creative bonus when both are packed.\nBringing 6 with 11 yields an extra 102 creative bonus when both are packed.\nBringing 6 with 12 yields an extra 103 creative bonus when both are packed.\nBringing 6 with 13 yields an extra 102 creative bonus when both are packed.\nBringing 6 with 14 yields an extra 104 creative bonus when both are packed.\nBringing 6 with 15 yields an extra 104 creative bonus when both are packed.\nBringing 7 with 8 yields an extra 102 creative bonus when both are packed.\nBringing 7 with 9 yields an extra 101 creative bonus when both are packed.\nBringing 7 with 10 yields an extra 101 creative bonus when both are packed.\nBringing 7 with 11 yields an extra 101 creative bonus when both are packed.\nBringing 7 with 12 yields an extra 102 creative bonus when both are packed.\nBringing 7 with 13 yields an extra 101 creative bonus when both are packed.\nBringing 7 with 14 yields an extra 102 creative bonus when both are packed.\nBringing 7 with 15 yields an extra 100 creative bonus when both are packed.\nBringing 8 with 9 yields an extra 102 creative bonus when both are packed.\nBringing 8 with 10 yields an extra 103 creative bonus when both are packed.\nBringing 8 with 11 yields an extra 101 creative bonus when both are packed.\nBringing 8 with 12 yields an extra 101 creative bonus when both are packed.\nBringing 8 with 13 yields an extra 102 creative bonus when both are packed.\nBringing 8 with 14 yields an extra 102 creative bonus when both are packed.\nBringing 8 with 15 yields an extra 103 creative bonus when both are packed.\nBringing 9 with 10 yields an extra 102 creative bonus when both are packed.\nBringing 9 with 11 yields an extra 102 creative bonus when both are packed.\nBringing 9 with 12 yields an extra 101 creative bonus when both are packed.\nBringing 9 with 13 yields an extra 102 creative bonus when both are packed.\nBringing 9 with 14 yields an extra 102 creative bonus when both are packed.\nBringing 9 with 15 yields an extra 100 creative bonus when both are packed.\nBringing 10 with 11 yields an extra 102 creative bonus when both are packed.\nBringing 10 with 12 yields an extra 104 creative bonus when both are packed.\nBringing 10 with 13 yields an extra 102 creative bonus when both are packed.\nBringing 10 with 14 yields an extra 102 creative bonus when both are packed.\nBringing 10 with 15 yields an extra 103 creative bonus when both are packed.\nBringing 11 with 12 yields an extra 104 creative bonus when both are packed.\nBringing 11 with 13 yields an extra 102 creative bonus when both are packed.\nBringing 11 with 14 yields an extra 103 creative bonus when both are packed.\nBringing 11 with 15 yields an extra 101 creative bonus when both are packed.\nBringing 12 with 13 yields an extra 102 creative bonus when both are packed.\nBringing 12 with 14 yields an extra 103 creative bonus when both are packed.\nBringing 12 with 15 yields an extra 102 creative bonus when both are packed.\nBringing 13 with 14 yields an extra 102 creative bonus when both are packed.\nBringing 13 with 15 yields an extra 102 creative bonus when both are packed.\nLet’s pack to maximize our total value while staying within the 876 limit.\n\nAlso, when you send back your pick, please use this simple JSON layout so it's easy to parse which kit bits you're bringing:\n\n{\n  \"solution\": [<selected_item_id>, <selected_item_id>, ...]\n}\n\nThe \"solution\" array should just list the item identifiers you want to pack — like ticking off the pieces you'll bring in your travel kit. It's just a sketch of the expected shape, not the actual answer itself.\n\nPlease use the identifiers exactly as they appear in the instance input — don't rename them or invent new labels. \n- for example: \"Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.\"",
    "instance": {
      "linear_coeffs": [
        104,
        103,
        105,
        103,
        100,
        105,
        102,
        103,
        103,
        104,
        103,
        102,
        102,
        102,
        103,
        101
      ],
      "quadratic_coeffs": [
        [
          101,
          101,
          101,
          101,
          100,
          100,
          100,
          100,
          101,
          101,
          102,
          101,
          102,
          101,
          104,
          102
        ],
        [
          0,
          102,
          100,
          100,
          101,
          100,
          100,
          100,
          100,
          100,
          103,
          103,
          101,
          103,
          103,
          103
        ],
        [
          0,
          0,
          103,
          102,
          102,
          100,
          102,
          102,
          102,
          103,
          101,
          102,
          102,
          100,
          100,
          102
        ],
        [
          0,
          0,
          0,
          101,
          101,
          100,
          102,
          102,
          101,
          100,
          100,
          100,
          101,
          100,
          100,
          101
        ],
        [
          0,
          0,
          0,
          0,
          102,
          102,
          100,
          102,
          100,
          100,
          101,
          101,
          101,
          100,
          101,
          101
        ],
        [
          0,
          0,
          0,
          0,
          0,
          104,
          104,
          104,
          100,
          100,
          100,
          100,
          100,
          100,
          100,
          100
        ],
        [
          0,
          0,
          0,
          0,
          0,
          0,
          103,
          103,
          104,
          102,
          102,
          102,
          103,
          102,
          104,
          104
        ],
        [
          0,
          0,
          0,
          0,
          0,
          0,
          0,
          102,
          102,
          101,
          101,
          101,
          102,
          101,
          102,
          100
        ],
        [
          0,
          0,
          0,
          0,
          0,
          0,
          0,
          0,
          102,
          102,
          103,
          101,
          101,
          102,
          102,
          103
        ],
        [
          0,
          0,
          0,
          0,
          0,
          0,
          0,
          0,
          0,
          101,
          102,
          102,
          101,
          102,
          102,
          100
        ],
        [
          0,
          0,
          0,
          0,
          0,
          0,
          0,
          0,
          0,
          0,
          103,
          102,
          104,
          102,
          102,
          103
        ],
        [
          0,
          0,
          0,
          0,
          0,
          0,
          0,
          0,
          0,
          0,
          0,
          104,
          104,
          102,
          103,
          101
        ],
        [
          0,
          0,
          0,
          0,
          0,
          0,
          0,
          0,
          0,
          0,
          0,
          0,
          102,
          102,
          103,
          102
        ],
        [
          0,
          0,
          0,
          0,
          0,
          0,
          0,
          0,
          0,
          0,
          0,
          0,
          0,
          103,
          102,
          102
        ],
        [
          0,
          0,
          0,
          0,
          0,
          0,
          0,
          0,
          0,
          0,
          0,
          0,
          0,
          0,
          0,
          0
        ],
        [
          0,
          0,
          0,
          0,
          0,
          0,
          0,
          0,
          0,
          0,
          0,
          0,
          0,
          0,
          0,
          0
        ]
      ],
      "weights": [
        101,
        100,
        102,
        101,
        102,
        105,
        103,
        104,
        100,
        102,
        105,
        100,
        104,
        102,
        102,
        101
      ],
      "capacity": 876,
      "solution": [
        0,
        1,
        4,
        6,
        8,
        10,
        12,
        14
      ],
      "obj": 3669.0,
      "problem_type": "QKP"
    },
    "solution": [
      0,
      1,
      4,
      6,
      8,
      10,
      12,
      14
    ],
    "obj": 3669.0,
    "instance_variant": {
      "problem_type": "QKP",
      "num_items": 16,
      "capacity": 876,
      "items": [
        0,
        1,
        2,
        3,
        4,
        5,
        6,
        7,
        8,
        9,
        10,
        11,
        12,
        13,
        14,
        15
      ],
      "linear_pairs": [
        {
          "item_id": 0,
          "linear_profit": 104
        },
        {
          "item_id": 1,
          "linear_profit": 103
        },
        {
          "item_id": 2,
          "linear_profit": 105
        },
        {
          "item_id": 3,
          "linear_profit": 103
        },
        {
          "item_id": 4,
          "linear_profit": 100
        },
        {
          "item_id": 5,
          "linear_profit": 105
        },
        {
          "item_id": 6,
          "linear_profit": 102
        },
        {
          "item_id": 7,
          "linear_profit": 103
        },
        {
          "item_id": 8,
          "linear_profit": 103
        },
        {
          "item_id": 9,
          "linear_profit": 104
        },
        {
          "item_id": 10,
          "linear_profit": 103
        },
        {
          "item_id": 11,
          "linear_profit": 102
        },
        {
          "item_id": 12,
          "linear_profit": 102
        },
        {
          "item_id": 13,
          "linear_profit": 102
        },
        {
          "item_id": 14,
          "linear_profit": 103
        },
        {
          "item_id": 15,
          "linear_profit": 101
        }
      ],
      "weight_pairs": [
        {
          "item_id": 0,
          "weight": 101
        },
        {
          "item_id": 1,
          "weight": 100
        },
        {
          "item_id": 2,
          "weight": 102
        },
        {
          "item_id": 3,
          "weight": 101
        },
        {
          "item_id": 4,
          "weight": 102
        },
        {
          "item_id": 5,
          "weight": 105
        },
        {
          "item_id": 6,
          "weight": 103
        },
        {
          "item_id": 7,
          "weight": 104
        },
        {
          "item_id": 8,
          "weight": 100
        },
        {
          "item_id": 9,
          "weight": 102
        },
        {
          "item_id": 10,
          "weight": 105
        },
        {
          "item_id": 11,
          "weight": 100
        },
        {
          "item_id": 12,
          "weight": 104
        },
        {
          "item_id": 13,
          "weight": 102
        },
        {
          "item_id": 14,
          "weight": 102
        },
        {
          "item_id": 15,
          "weight": 101
        }
      ],
      "quadratic_pairs": [
        {
          "item_i_id": 0,
          "item_j_id": 0,
          "quadratic_profit": 101
        },
        {
          "item_i_id": 0,
          "item_j_id": 1,
          "quadratic_profit": 101
        },
        {
          "item_i_id": 0,
          "item_j_id": 2,
          "quadratic_profit": 101
        },
        {
          "item_i_id": 0,
          "item_j_id": 3,
          "quadratic_profit": 101
        },
        {
          "item_i_id": 0,
          "item_j_id": 4,
          "quadratic_profit": 100
        },
        {
          "item_i_id": 0,
          "item_j_id": 5,
          "quadratic_profit": 100
        },
        {
          "item_i_id": 0,
          "item_j_id": 6,
          "quadratic_profit": 100
        },
        {
          "item_i_id": 0,
          "item_j_id": 7,
          "quadratic_profit": 100
        },
        {
          "item_i_id": 0,
          "item_j_id": 8,
          "quadratic_profit": 101
        },
        {
          "item_i_id": 0,
          "item_j_id": 9,
          "quadratic_profit": 101
        },
        {
          "item_i_id": 0,
          "item_j_id": 10,
          "quadratic_profit": 102
        },
        {
          "item_i_id": 0,
          "item_j_id": 11,
          "quadratic_profit": 101
        },
        {
          "item_i_id": 0,
          "item_j_id": 12,
          "quadratic_profit": 102
        },
        {
          "item_i_id": 0,
          "item_j_id": 13,
          "quadratic_profit": 101
        },
        {
          "item_i_id": 0,
          "item_j_id": 14,
          "quadratic_profit": 104
        },
        {
          "item_i_id": 0,
          "item_j_id": 15,
          "quadratic_profit": 102
        },
        {
          "item_i_id": 1,
          "item_j_id": 1,
          "quadratic_profit": 102
        },
        {
          "item_i_id": 1,
          "item_j_id": 2,
          "quadratic_profit": 100
        },
        {
          "item_i_id": 1,
          "item_j_id": 3,
          "quadratic_profit": 100
        },
        {
          "item_i_id": 1,
          "item_j_id": 4,
          "quadratic_profit": 101
        },
        {
          "item_i_id": 1,
          "item_j_id": 5,
          "quadratic_profit": 100
        },
        {
          "item_i_id": 1,
          "item_j_id": 6,
          "quadratic_profit": 100
        },
        {
          "item_i_id": 1,
          "item_j_id": 7,
          "quadratic_profit": 100
        },
        {
          "item_i_id": 1,
          "item_j_id": 8,
          "quadratic_profit": 100
        },
        {
          "item_i_id": 1,
          "item_j_id": 9,
          "quadratic_profit": 100
        },
        {
          "item_i_id": 1,
          "item_j_id": 10,
          "quadratic_profit": 103
        },
        {
          "item_i_id": 1,
          "item_j_id": 11,
          "quadratic_profit": 103
        },
        {
          "item_i_id": 1,
          "item_j_id": 12,
          "quadratic_profit": 101
        },
        {
          "item_i_id": 1,
          "item_j_id": 13,
          "quadratic_profit": 103
        },
        {
          "item_i_id": 1,
          "item_j_id": 14,
          "quadratic_profit": 103
        },
        {
          "item_i_id": 1,
          "item_j_id": 15,
          "quadratic_profit": 103
        },
        {
          "item_i_id": 2,
          "item_j_id": 2,
          "quadratic_profit": 103
        },
        {
          "item_i_id": 2,
          "item_j_id": 3,
          "quadratic_profit": 102
        },
        {
          "item_i_id": 2,
          "item_j_id": 4,
          "quadratic_profit": 102
        },
        {
          "item_i_id": 2,
          "item_j_id": 5,
          "quadratic_profit": 100
        },
        {
          "item_i_id": 2,
          "item_j_id": 6,
          "quadratic_profit": 102
        },
        {
          "item_i_id": 2,
          "item_j_id": 7,
          "quadratic_profit": 102
        },
        {
          "item_i_id": 2,
          "item_j_id": 8,
          "quadratic_profit": 102
        },
        {
          "item_i_id": 2,
          "item_j_id": 9,
          "quadratic_profit": 103
        },
        {
          "item_i_id": 2,
          "item_j_id": 10,
          "quadratic_profit": 101
        },
        {
          "item_i_id": 2,
          "item_j_id": 11,
          "quadratic_profit": 102
        },
        {
          "item_i_id": 2,
          "item_j_id": 12,
          "quadratic_profit": 102
        },
        {
          "item_i_id": 2,
          "item_j_id": 13,
          "quadratic_profit": 100
        },
        {
          "item_i_id": 2,
          "item_j_id": 14,
          "quadratic_profit": 100
        },
        {
          "item_i_id": 2,
          "item_j_id": 15,
          "quadratic_profit": 102
        },
        {
          "item_i_id": 3,
          "item_j_id": 3,
          "quadratic_profit": 101
        },
        {
          "item_i_id": 3,
          "item_j_id": 4,
          "quadratic_profit": 101
        },
        {
          "item_i_id": 3,
          "item_j_id": 5,
          "quadratic_profit": 100
        },
        {
          "item_i_id": 3,
          "item_j_id": 6,
          "quadratic_profit": 102
        },
        {
          "item_i_id": 3,
          "item_j_id": 7,
          "quadratic_profit": 102
        },
        {
          "item_i_id": 3,
          "item_j_id": 8,
          "quadratic_profit": 101
        },
        {
          "item_i_id": 3,
          "item_j_id": 9,
          "quadratic_profit": 100
        },
        {
          "item_i_id": 3,
          "item_j_id": 10,
          "quadratic_profit": 100
        },
        {
          "item_i_id": 3,
          "item_j_id": 11,
          "quadratic_profit": 100
        },
        {
          "item_i_id": 3,
          "item_j_id": 12,
          "quadratic_profit": 101
        },
        {
          "item_i_id": 3,
          "item_j_id": 13,
          "quadratic_profit": 100
        },
        {
          "item_i_id": 3,
          "item_j_id": 14,
          "quadratic_profit": 100
        },
        {
          "item_i_id": 3,
          "item_j_id": 15,
          "quadratic_profit": 101
        },
        {
          "item_i_id": 4,
          "item_j_id": 4,
          "quadratic_profit": 102
        },
        {
          "item_i_id": 4,
          "item_j_id": 5,
          "quadratic_profit": 102
        },
        {
          "item_i_id": 4,
          "item_j_id": 6,
          "quadratic_profit": 100
        },
        {
          "item_i_id": 4,
          "item_j_id": 7,
          "quadratic_profit": 102
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    "task_name": "QKP",
    "problem_type": "QKP",
    "instruction": "Around the neighborhood a landscaper is planning a compact front yard and sorting through which plants to use. Each shrub or tree takes up a certain area and brings a base curb-appeal score, and certain pairs have little synergy that adds to the overall look if both are planted. The choice is which plants to place so the total area stays within the plot’s limit (each plant chosen once), and the yard’s total curb appeal comes from adding each chosen plant’s own appeal plus any extra points for every planted pair that has synergy. The concrete list of plants and their stats is provided below.\n\nThey are considering 19 plants (0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18) and must keep the total planted area within 4845.\n\n| plant_id | base_appeal |\n|---|---|\n| 0 | 694 |\n| 1 | 248 |\n| 2 | 351 |\n| 3 | 737 |\n| 4 | 776 |\n| 5 | 560 |\n| 6 | 940 |\n| 7 | 484 |\n| 8 | 936 |\n| 9 | 844 |\n| 10 | 658 |\n| 11 | 231 |\n| 12 | 880 |\n| 13 | 625 |\n| 14 | 421 |\n| 15 | 206 |\n| 16 | 691 |\n| 17 | 636 |\n| 18 | 818 |\n\n| plant_id | area_required |\n|---|---|\n| 0 | 604 |\n| 1 | 158 |\n| 2 | 261 |\n| 3 | 647 |\n| 4 | 686 |\n| 5 | 470 |\n| 6 | 850 |\n| 7 | 394 |\n| 8 | 846 |\n| 9 | 754 |\n| 10 | 568 |\n| 11 | 141 |\n| 12 | 790 |\n| 13 | 535 |\n| 14 | 331 |\n| 15 | 116 |\n| 16 | 601 |\n| 17 | 546 |\n| 18 | 728 |\n\n| plant_i_id | plant_j_id | pair_synergy_appeal |\n|---|---|---|\n| 0 | 1 | 653 |\n| 0 | 2 | 633 |\n| 0 | 3 | 661 |\n| 0 | 4 | 348 |\n| 0 | 5 | 724 |\n| 0 | 6 | 674 |\n| 0 | 7 | 716 |\n| 0 | 8 | 633 |\n| 0 | 9 | 502 |\n| 0 | 10 | 590 |\n| 0 | 11 | 368 |\n| 0 | 12 | 656 |\n| 0 | 13 | 345 |\n| 0 | 14 | 730 |\n| 0 | 15 | 266 |\n| 0 | 16 | 395 |\n| 0 | 17 | 471 |\n| 0 | 18 | 270 |\n| 1 | 2 | 178 |\n| 1 | 3 | 178 |\n| 1 | 4 | 370 |\n| 1 | 5 | 148 |\n| 1 | 6 | 228 |\n| 1 | 7 | 267 |\n| 1 | 8 | 299 |\n| 1 | 9 | 254 |\n| 1 | 10 | 252 |\n| 1 | 11 | 279 |\n| 1 | 12 | 326 |\n| 1 | 13 | 308 |\n| 1 | 14 | 305 |\n| 1 | 15 | 293 |\n| 1 | 16 | 339 |\n| 1 | 17 | 739 |\n| 1 | 18 | 523 |\n| 2 | 3 | 191 |\n| 2 | 4 | 471 |\n| 2 | 5 | 237 |\n| 2 | 6 | 344 |\n| 2 | 7 | 330 |\n| 2 | 8 | 318 |\n| 2 | 9 | 471 |\n| 2 | 10 | 454 |\n| 2 | 11 | 398 |\n| 2 | 12 | 326 |\n| 2 | 13 | 175 |\n| 2 | 14 | 348 |\n| 2 | 15 | 259 |\n| 2 | 16 | 309 |\n| 2 | 17 | 329 |\n| 2 | 18 | 636 |\n| 3 | 4 | 574 |\n| 3 | 5 | 741 |\n| 3 | 6 | 384 |\n| 3 | 7 | 679 |\n| 3 | 8 | 418 |\n| 3 | 9 | 513 |\n| 3 | 10 | 588 |\n| 3 | 11 | 603 |\n| 3 | 12 | 276 |\n| 3 | 13 | 309 |\n| 3 | 14 | 763 |\n| 3 | 15 | 666 |\n| 3 | 16 | 548 |\n| 3 | 17 | 763 |\n| 3 | 18 | 535 |\n| 4 | 5 | 311 |\n| 4 | 6 | 620 |\n| 4 | 7 | 679 |\n| 4 | 8 | 691 |\n| 4 | 9 | 697 |\n| 4 | 10 | 531 |\n| 4 | 11 | 368 |\n| 4 | 12 | 612 |\n| 4 | 13 | 632 |\n| 4 | 14 | 551 |\n| 4 | 15 | 536 |\n| 4 | 16 | 394 |\n| 4 | 17 | 558 |\n| 4 | 18 | 427 |\n| 5 | 6 | 534 |\n| 5 | 7 | 356 |\n| 5 | 8 | 437 |\n| 5 | 9 | 273 |\n| 5 | 10 | 394 |\n| 5 | 11 | 531 |\n| 5 | 12 | 466 |\n| 5 | 13 | 692 |\n| 5 | 14 | 673 |\n| 5 | 15 | 570 |\n| 5 | 16 | 793 |\n| 5 | 17 | 556 |\n| 5 | 18 | 793 |\n| 6 | 7 | 530 |\n| 6 | 8 | 314 |\n| 6 | 9 | 691 |\n| 6 | 10 | 681 |\n| 6 | 11 | 706 |\n| 6 | 12 | 427 |\n| 6 | 13 | 430 |\n| 6 | 14 | 413 |\n| 6 | 15 | 220 |\n| 6 | 16 | 362 |\n| 6 | 17 | 388 |\n| 6 | 18 | 295 |\n| 7 | 8 | 279 |\n| 7 | 9 | 589 |\n| 7 | 10 | 319 |\n| 7 | 11 | 518 |\n| 7 | 12 | 527 |\n| 7 | 13 | 373 |\n| 7 | 14 | 609 |\n| 7 | 15 | 641 |\n| 7 | 16 | 659 |\n| 7 | 17 | 279 |\n| 7 | 18 | 671 |\n| 8 | 9 | 315 |\n| 8 | 10 | 357 |\n| 8 | 11 | 610 |\n| 8 | 12 | 423 |\n| 8 | 13 | 543 |\n| 8 | 14 | 553 |\n| 8 | 15 | 558 |\n| 8 | 16 | 237 |\n| 8 | 17 | 568 |\n| 8 | 18 | 294 |\n| 9 | 10 | 526 |\n| 9 | 11 | 371 |\n| 9 | 12 | 216 |\n| 9 | 13 | 484 |\n| 9 | 14 | 366 |\n| 9 | 15 | 205 |\n| 9 | 16 | 493 |\n| 9 | 17 | 256 |\n| 9 | 18 | 197 |\n| 10 | 11 | 306 |\n| 10 | 12 | 546 |\n| 10 | 13 | 287 |\n| 10 | 14 | 607 |\n| 10 | 15 | 222 |\n| 10 | 16 | 329 |\n| 10 | 17 | 392 |\n| 10 | 18 | 240 |\n| 11 | 12 | 437 |\n| 11 | 13 | 460 |\n| 11 | 14 | 573 |\n| 11 | 15 | 479 |\n| 11 | 16 | 474 |\n| 11 | 17 | 330 |\n| 11 | 18 | 252 |\n| 12 | 13 | 399 |\n| 12 | 14 | 301 |\n| 12 | 15 | 169 |\n| 12 | 16 | 407 |\n| 12 | 17 | 211 |\n| 12 | 18 | 162 |\n| 13 | 14 | 132 |\n| 13 | 15 | 206 |\n| 13 | 16 | 395 |\n| 13 | 17 | 286 |\n| 13 | 18 | 163 |\n| 14 | 15 | 172 |\n| 14 | 16 | 595 |\n| 14 | 17 | 573 |\n| 14 | 18 | 593 |\n| 15 | 16 | 293 |\n| 15 | 17 | 503 |\n| 15 | 18 | 287 |\n\nThe landscaper will select from the 19 options to maximize total curb appeal while staying within 4845.\n\nOh, and when you're ready to give the final pick, a little JSON snippet like the one below is handy — just a relaxed, predictable shape so whoever reads it knows where to look.\n\n{\n  \"solution\": [<selected_plant_id>, <selected_plant_id>, ...]\n}\n\nThink of \"solution\" as the list of plant IDs you decide to put in the yard. The placeholders inside the brackets stand in for the exact plant identifiers from the instance input; this block is just a sketch of the expected shape, not the actual answer.\n\nPlease make sure to use the identifiers exactly as they appear in the instance input — don't rename them or add new labels. \n\nFor example: Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.",
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    "instruction": "Recently a small dinner was planned and the host had to choose appetizers carefully. Every choice uses a bit of pantry/fridge space and gives a base amount of satisfaction, and a few particular combinations add extra joy when they’re paired. The challenge is to select which appetizers will fit into the available storage space without repeating items, and to get the highest possible total happiness, counted as the sum of the base satisfactions plus any bonuses for pairs that are both selected. The specific dishes and their space/satisfaction/pairing numbers are shown below.\n\n# num_appetizers=24\n# storage_capacity=78227\n# appetizer_ids=A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V, W, X\n\nappetizer_id,base_satisfaction\nA,6419\nB,6692\nC,6123\nD,1432\nE,6916\nF,5357\nG,6413\nH,2766\nI,2707\nJ,3896\nK,2774\nL,3552\nM,1107\nN,2613\nO,2572\nP,911\nQ,6983\nR,2413\nS,2388\nT,6800\nU,3568\nV,2642\nW,6948\nX,5802\n\nappetizer_id,storage_space\nA,5769\nB,6042\nC,5473\nD,782\nE,6266\nF,4707\nG,5763\nH,2116\nI,2057\nJ,3246\nK,2124\nL,2902\nM,457\nN,1963\nO,1922\nP,261\nQ,6333\nR,1763\nS,1738\nT,6150\nU,2918\nV,1992\nW,6298\nX,5152\n\nappetizer_i_id,appetizer_j_id,pairing_bonus\nA,B,3911\nA,C,4946\nA,D,3051\nA,E,2002\nA,G,2116\nA,H,5769\nA,I,1122\nA,J,4889\nA,K,4985\nA,L,5769\nA,M,1080\nA,N,2373\nA,O,5769\nA,P,4599\nA,Q,3872\nA,R,1999\nA,S,2080\nA,T,5267\nA,U,576\nA,V,1577\nB,D,3729\nB,E,2283\nB,F,5018\nB,G,320\nB,H,4149\nB,I,381\nB,J,1811\nB,K,1668\nB,M,6042\nB,N,2550\nB,O,3779\nB,P,1859\nB,Q,2387\nB,R,4888\nB,S,5152\nB,T,1727\nB,U,1727\nB,W,1727\nB,X,1727\nC,E,575\nC,F,1863\nC,G,2737\nC,H,1669\nC,I,2061\nC,K,5473\nC,L,1741\nC,N,2251\nC,O,3971\nC,P,5151\nC,Q,2425\nC,R,1738\nC,S,5276\nC,T,782\nC,U,2913\nC,V,2984\nC,W,2666\nC,X,4691\nD,J,782\nD,L,782\nD,M,261\nD,N,782\nD,O,782\nD,P,782\nD,Q,782\nD,R,782\nD,S,4769\nD,T,4769\nD,U,419\nD,V,2102\nD,W,575\nE,F,2057\nE,G,5456\nE,I,2229\nE,J,2498\nE,K,2085\nE,M,3677\nE,N,4370\nE,P,3444\nE,Q,5152\nE,S,3116\nE,T,3116\nE,U,3116\nE,V,3116\nE,X,1049\nF,G,2704\nF,H,3797\nF,I,4707\nF,J,4707\nF,K,1261\nF,L,3971\nF,N,4212\nF,P,3754\nF,Q,3261\nF,R,3261\nF,T,3261\nF,U,2974\nF,V,2124\nF,W,381\nF,X,3261\nG,H,457\nG,I,1659\nG,K,3677\nG,M,1497\nG,N,5763\nG,O,1763\nG,P,2317\nG,Q,3008\nG,R,3008\nG,T,3008\nG,U,3008\nG,V,2383\nG,X,3008\nH,I,1261\nH,K,2116\nH,O,1999\nH,P,2116\nH,Q,951\nH,R,951\nH,S,951\nH,U,951\nH,V,951\nH,W,177\nH,X,951\nI,J,2057\nI,K,2057\nI,L,2057\nI,M,985\nI,O,2057\nI,P,2057\nI,R,734\nI,S,3820\nI,T,961\nI,U,172\nI,V,3971\nI,X,186\nJ,K,455\nJ,L,3246\nJ,M,3246\nJ,O,3246\nJ,P,1765\nJ,Q,1261\nJ,R,261\nJ,S,1765\nJ,T,1765\nJ,U,1765\nJ,W,1093\nJ,X,1765\nK,M,2124\nK,N,1577\nK,P,2381\nK,T,2381\nK,X,2381\nL,M,576\nL,N,1617\nL,Q,3797\nL,R,3797\nL,V,2425\nL,W,1738\nL,X,3797\nM,P,177\nM,Q,139\nM,V,1285\nM,W,793\nM,X,1442\nN,P,1571\nN,Q,4773\nN,R,279\nN,U,3475\nN,V,4773\nN,W,2207\nN,X,1497\nO,P,1763\nO,Q,3827\nO,S,3827\nO,T,3827\nO,U,3163\nO,V,2333\nO,W,2440\nO,X,3827\nP,Q,3156\nP,R,1497\nP,S,2027\nP,T,3156\nP,U,985\nP,X,3156\nQ,R,985\nQ,T,985\nQ,U,139\nQ,V,2169\nQ,W,3872\nQ,X,503\nR,S,1999\nR,T,1372\nR,U,1999\nR,V,1999\nR,X,1999\n\nIf you want to send me your pick, just drop it in this simple JSON shape so it's easy to read:\n\n{\n  \"solution\": [<selected_appetizer_id>, <selected_appetizer_id>, ...]\n}\n\nHere \"solution\" is just the list of appetizer IDs you’re choosing for the spread — one entry per selected dish. Think of it like filling out a short form: put each dish’s identifier in the array, and that’s your selection. This block is only a sketch of the shape I expect, not the actual answer.\n\nPlease make sure to use the identifiers exactly as they appear in the instance input — don’t rename them or invent new labels.  \n- \"Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.\"",
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        },
        {
          "item_i_id": "O",
          "item_j_id": "O",
          "quadratic_profit": 2162
        },
        {
          "item_i_id": "O",
          "item_j_id": "P",
          "quadratic_profit": 1763
        },
        {
          "item_i_id": "O",
          "item_j_id": "Q",
          "quadratic_profit": 3827
        },
        {
          "item_i_id": "O",
          "item_j_id": "S",
          "quadratic_profit": 3827
        },
        {
          "item_i_id": "O",
          "item_j_id": "T",
          "quadratic_profit": 3827
        },
        {
          "item_i_id": "O",
          "item_j_id": "U",
          "quadratic_profit": 3163
        },
        {
          "item_i_id": "O",
          "item_j_id": "V",
          "quadratic_profit": 2333
        },
        {
          "item_i_id": "O",
          "item_j_id": "W",
          "quadratic_profit": 2440
        },
        {
          "item_i_id": "O",
          "item_j_id": "X",
          "quadratic_profit": 3827
        },
        {
          "item_i_id": "P",
          "item_j_id": "P",
          "quadratic_profit": 4213
        },
        {
          "item_i_id": "P",
          "item_j_id": "Q",
          "quadratic_profit": 3156
        },
        {
          "item_i_id": "P",
          "item_j_id": "R",
          "quadratic_profit": 1497
        },
        {
          "item_i_id": "P",
          "item_j_id": "S",
          "quadratic_profit": 2027
        },
        {
          "item_i_id": "P",
          "item_j_id": "T",
          "quadratic_profit": 3156
        },
        {
          "item_i_id": "P",
          "item_j_id": "U",
          "quadratic_profit": 985
        },
        {
          "item_i_id": "P",
          "item_j_id": "X",
          "quadratic_profit": 3156
        },
        {
          "item_i_id": "Q",
          "item_j_id": "Q",
          "quadratic_profit": 985
        },
        {
          "item_i_id": "Q",
          "item_j_id": "R",
          "quadratic_profit": 985
        },
        {
          "item_i_id": "Q",
          "item_j_id": "T",
          "quadratic_profit": 985
        },
        {
          "item_i_id": "Q",
          "item_j_id": "U",
          "quadratic_profit": 139
        },
        {
          "item_i_id": "Q",
          "item_j_id": "V",
          "quadratic_profit": 2169
        },
        {
          "item_i_id": "Q",
          "item_j_id": "W",
          "quadratic_profit": 3872
        },
        {
          "item_i_id": "Q",
          "item_j_id": "X",
          "quadratic_profit": 503
        },
        {
          "item_i_id": "R",
          "item_j_id": "R",
          "quadratic_profit": 1629
        },
        {
          "item_i_id": "R",
          "item_j_id": "S",
          "quadratic_profit": 1999
        },
        {
          "item_i_id": "R",
          "item_j_id": "T",
          "quadratic_profit": 1372
        },
        {
          "item_i_id": "R",
          "item_j_id": "U",
          "quadratic_profit": 1999
        },
        {
          "item_i_id": "R",
          "item_j_id": "V",
          "quadratic_profit": 1999
        },
        {
          "item_i_id": "R",
          "item_j_id": "X",
          "quadratic_profit": 1999
        }
      ]
    },
    "solution_variant": [
      "A",
      "B",
      "C",
      "D",
      "E",
      "F",
      "G",
      "H",
      "I",
      "J",
      "K",
      "L",
      "M",
      "N",
      "O",
      "P",
      "Q",
      "R",
      "S",
      "T",
      "U",
      "V",
      "X"
    ],
    "context_index": 34,
    "input_format": "csv",
    "input_index_base": "names"
  },
  {
    "task_name": "QKP",
    "problem_type": "QKP",
    "instruction": "We’re helping a student prep for back-to-back classes and need to pick which textbooks and materials to carry in one backpack. Every item adds a base study value and has a weight, and certain combinations (like a textbook and its workbook) give an extra perk if both are brought. The trick is to get the highest possible study payoff by adding up the values of the chosen items and including any extra bonuses for compatible pairs, all without exceeding the backpack’s weight capacity and without duplicating items. The concrete details are listed below.\n\nHere are those details for the 24 materials: ids 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, backpack weight limit 4379.\n\n| material_id | base_study_value |\n|---|---|\n| 1 | 153 |\n| 2 | 150 |\n| 3 | 319 |\n| 4 | 286 |\n| 5 | 323 |\n| 6 | 238 |\n| 7 | 255 |\n| 8 | 142 |\n| 9 | 210 |\n| 10 | 246 |\n| 11 | 180 |\n| 12 | 247 |\n| 13 | 316 |\n| 14 | 206 |\n| 15 | 322 |\n| 16 | 201 |\n| 17 | 307 |\n| 18 | 266 |\n| 19 | 234 |\n| 20 | 321 |\n| 21 | 236 |\n| 22 | 140 |\n| 23 | 288 |\n| 24 | 270 |\n\n| material_id | material_weight |\n|---|---|\n| 1 | 123 |\n| 2 | 120 |\n| 3 | 289 |\n| 4 | 256 |\n| 5 | 293 |\n| 6 | 208 |\n| 7 | 225 |\n| 8 | 112 |\n| 9 | 180 |\n| 10 | 216 |\n| 11 | 150 |\n| 12 | 217 |\n| 13 | 286 |\n| 14 | 176 |\n| 15 | 292 |\n| 16 | 171 |\n| 17 | 277 |\n| 18 | 236 |\n| 19 | 204 |\n| 20 | 291 |\n| 21 | 206 |\n| 22 | 110 |\n| 23 | 258 |\n| 24 | 240 |\n\n| material_i_id | material_j_id | pair_study_bonus |\n|---|---|---|\n| 1 | 4 | 220 |\n| 1 | 5 | 198 |\n| 1 | 6 | 119 |\n| 1 | 8 | 212 |\n| 1 | 13 | 298 |\n| 1 | 15 | 158 |\n| 1 | 16 | 290 |\n| 1 | 19 | 191 |\n| 1 | 20 | 255 |\n| 1 | 21 | 175 |\n| 1 | 22 | 125 |\n| 1 | 23 | 245 |\n| 2 | 3 | 152 |\n| 2 | 4 | 278 |\n| 2 | 9 | 129 |\n| 2 | 11 | 298 |\n| 2 | 12 | 105 |\n| 2 | 15 | 201 |\n| 2 | 16 | 245 |\n| 2 | 19 | 201 |\n| 2 | 21 | 145 |\n| 2 | 23 | 119 |\n| 2 | 24 | 174 |\n| 3 | 5 | 113 |\n| 3 | 7 | 178 |\n| 3 | 9 | 234 |\n| 3 | 10 | 173 |\n| 3 | 12 | 149 |\n| 3 | 14 | 220 |\n| 3 | 15 | 132 |\n| 3 | 16 | 272 |\n| 3 | 19 | 234 |\n| 3 | 20 | 199 |\n| 3 | 21 | 206 |\n| 3 | 22 | 118 |\n| 3 | 23 | 209 |\n| 4 | 8 | 260 |\n| 4 | 9 | 262 |\n| 4 | 11 | 298 |\n| 4 | 13 | 164 |\n| 4 | 14 | 152 |\n| 4 | 16 | 148 |\n| 4 | 20 | 105 |\n| 4 | 22 | 194 |\n| 4 | 24 | 247 |\n| 5 | 6 | 231 |\n| 5 | 9 | 294 |\n| 5 | 12 | 121 |\n| 5 | 13 | 192 |\n| 5 | 15 | 169 |\n| 5 | 16 | 179 |\n| 5 | 17 | 238 |\n| 5 | 20 | 220 |\n| 5 | 21 | 124 |\n| 5 | 22 | 284 |\n| 5 | 23 | 224 |\n| 6 | 9 | 141 |\n| 6 | 10 | 169 |\n| 6 | 12 | 191 |\n| 6 | 13 | 236 |\n| 6 | 14 | 150 |\n| 6 | 15 | 228 |\n| 6 | 18 | 171 |\n| 6 | 20 | 241 |\n| 6 | 21 | 117 |\n| 6 | 22 | 220 |\n| 6 | 23 | 289 |\n| 6 | 24 | 139 |\n| 7 | 8 | 175 |\n| 7 | 9 | 269 |\n| 7 | 10 | 249 |\n| 7 | 11 | 158 |\n| 7 | 12 | 202 |\n| 7 | 14 | 175 |\n| 7 | 18 | 266 |\n| 7 | 21 | 285 |\n| 7 | 22 | 242 |\n| 7 | 24 | 106 |\n| 8 | 10 | 225 |\n| 8 | 12 | 238 |\n| 8 | 16 | 204 |\n| 8 | 20 | 138 |\n| 8 | 24 | 193 |\n| 9 | 13 | 244 |\n| 9 | 14 | 124 |\n| 9 | 15 | 232 |\n| 9 | 19 | 238 |\n| 9 | 20 | 248 |\n| 9 | 22 | 227 |\n| 9 | 23 | 176 |\n| 10 | 17 | 201 |\n| 10 | 18 | 104 |\n| 10 | 20 | 207 |\n| 10 | 23 | 226 |\n| 10 | 24 | 171 |\n| 11 | 15 | 258 |\n| 11 | 17 | 134 |\n| 11 | 18 | 264 |\n| 11 | 22 | 172 |\n| 12 | 15 | 150 |\n| 12 | 20 | 135 |\n| 12 | 21 | 135 |\n| 12 | 24 | 203 |\n| 13 | 17 | 212 |\n| 13 | 18 | 150 |\n| 13 | 20 | 130 |\n| 13 | 21 | 164 |\n| 13 | 23 | 141 |\n| 13 | 24 | 155 |\n| 14 | 15 | 199 |\n| 14 | 16 | 275 |\n| 14 | 19 | 119 |\n| 14 | 22 | 211 |\n| 14 | 24 | 257 |\n| 15 | 16 | 296 |\n| 15 | 17 | 242 |\n| 15 | 18 | 239 |\n| 15 | 20 | 114 |\n| 15 | 21 | 265 |\n| 15 | 22 | 261 |\n| 15 | 23 | 174 |\n| 15 | 24 | 299 |\n| 16 | 17 | 211 |\n| 16 | 18 | 274 |\n| 16 | 19 | 116 |\n| 16 | 20 | 178 |\n| 16 | 21 | 166 |\n| 16 | 23 | 118 |\n| 16 | 24 | 279 |\n| 17 | 18 | 106 |\n| 17 | 19 | 108 |\n| 17 | 23 | 149 |\n| 17 | 24 | 246 |\n| 18 | 19 | 119 |\n| 18 | 21 | 218 |\n| 18 | 23 | 228 |\n| 19 | 21 | 126 |\n| 19 | 22 | 257 |\n| 19 | 23 | 210 |\n| 20 | 23 | 133 |\n| 20 | 24 | 104 |\n| 21 | 22 | 274 |\n| 21 | 23 | 114 |\n| 22 | 23 | 297 |\n| 23 | 24 | 223 |\n\nWe'll pack to maximize study payoff while staying within the 4379 limit.\n\nWhen you send your pick, just drop it in a tiny JSON snippet so it's easy to read and check. Something casual like this works:\n\n{\n  \"solution\": [<selected_book_id>, <selected_book_id>, ...]\n}\n\nThis just means: \"solution\" is the list of items (books/materials) you're planning to bring — put the identifiers for each selected item inside the array. Think of it like filling out a short form: list the exact IDs of the things you want in your backpack. The JSON above is only a sketch of the shape we expect, not the actual answer.\n\nPlease be sure to use the item identifiers exactly as they appear in the instance input — no renaming and no new labels. Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.",
    "instance": {
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        153,
        150,
        319,
        286,
        323,
        238,
        255,
        142,
        210,
        246,
        180,
        247,
        316,
        206,
        322,
        201,
        307,
        266,
        234,
        321,
        236,
        140,
        288,
        270
      ],
      "quadratic_coeffs": [
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          0
        ],
        [
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          0,
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          105,
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          245,
          0,
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        ],
        [
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          0,
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          178,
          0,
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          0,
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          118,
          209,
          0
        ],
        [
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          0,
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        [
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        [
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        ],
        [
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        [
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        ],
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        ],
        [
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        ]
      ],
      "weights": [
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        120,
        289,
        256,
        293,
        208,
        225,
        112,
        180,
        216,
        150,
        217,
        286,
        176,
        292,
        171,
        277,
        236,
        204,
        291,
        206,
        110,
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      ],
      "capacity": 4379,
      "solution": [
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        11,
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        18,
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        20,
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      ],
      "obj": 28241.0,
      "problem_type": "QKP"
    },
    "solution": [
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      21,
      22,
      23
    ],
    "obj": 28241.0,
    "instance_variant": {
      "problem_type": "QKP",
      "num_items": 24,
      "capacity": 4379,
      "items": [
        1,
        2,
        3,
        4,
        5,
        6,
        7,
        8,
        9,
        10,
        11,
        12,
        13,
        14,
        15,
        16,
        17,
        18,
        19,
        20,
        21,
        22,
        23,
        24
      ],
      "linear_pairs": [
        {
          "item_id": 1,
          "linear_profit": 153
        },
        {
          "item_id": 2,
          "linear_profit": 150
        },
        {
          "item_id": 3,
          "linear_profit": 319
        },
        {
          "item_id": 4,
          "linear_profit": 286
        },
        {
          "item_id": 5,
          "linear_profit": 323
        },
        {
          "item_id": 6,
          "linear_profit": 238
        },
        {
          "item_id": 7,
          "linear_profit": 255
        },
        {
          "item_id": 8,
          "linear_profit": 142
        },
        {
          "item_id": 9,
          "linear_profit": 210
        },
        {
          "item_id": 10,
          "linear_profit": 246
        },
        {
          "item_id": 11,
          "linear_profit": 180
        },
        {
          "item_id": 12,
          "linear_profit": 247
        },
        {
          "item_id": 13,
          "linear_profit": 316
        },
        {
          "item_id": 14,
          "linear_profit": 206
        },
        {
          "item_id": 15,
          "linear_profit": 322
        },
        {
          "item_id": 16,
          "linear_profit": 201
        },
        {
          "item_id": 17,
          "linear_profit": 307
        },
        {
          "item_id": 18,
          "linear_profit": 266
        },
        {
          "item_id": 19,
          "linear_profit": 234
        },
        {
          "item_id": 20,
          "linear_profit": 321
        },
        {
          "item_id": 21,
          "linear_profit": 236
        },
        {
          "item_id": 22,
          "linear_profit": 140
        },
        {
          "item_id": 23,
          "linear_profit": 288
        },
        {
          "item_id": 24,
          "linear_profit": 270
        }
      ],
      "weight_pairs": [
        {
          "item_id": 1,
          "weight": 123
        },
        {
          "item_id": 2,
          "weight": 120
        },
        {
          "item_id": 3,
          "weight": 289
        },
        {
          "item_id": 4,
          "weight": 256
        },
        {
          "item_id": 5,
          "weight": 293
        },
        {
          "item_id": 6,
          "weight": 208
        },
        {
          "item_id": 7,
          "weight": 225
        },
        {
          "item_id": 8,
          "weight": 112
        },
        {
          "item_id": 9,
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    "problem_type": "QKP",
    "instruction": "We’re putting together a single promotional shelf and deciding which product lines to feature. Every item has a weight and an estimated sales figure, and certain combinations give a little bonus because they lift each other’s sales. The trick is to decide which items to put on that shelf so the combined weight doesn’t exceed what the shelf can hold, you only pick each product once (either it’s on the shelf or it isn’t), and the overall revenue — calculated by adding each chosen item’s expected sales plus any pairwise bonus sales — comes out as high as possible. The concrete details and numbers are listed below.\n\nWe have 20 product lines (A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T) and a shelf capacity of 70207.\nWe estimate 3913 in base sales if we stock A.\nWe estimate 2255 in base sales if we stock B.\nWe estimate 1503 in base sales if we stock C.\nWe estimate 5766 in base sales if we stock D.\nWe estimate 4188 in base sales if we stock E.\nWe estimate 2000 in base sales if we stock F.\nWe estimate 3309 in base sales if we stock G.\nWe estimate 4455 in base sales if we stock H.\nWe estimate 1277 in base sales if we stock I.\nWe estimate 1142 in base sales if we stock J.\nWe estimate 3374 in base sales if we stock K.\nWe estimate 5308 in base sales if we stock L.\nWe estimate 2582 in base sales if we stock M.\nWe estimate 4728 in base sales if we stock N.\nWe estimate 1526 in base sales if we stock O.\nWe estimate 1683 in base sales if we stock P.\nWe estimate 2971 in base sales if we stock Q.\nWe estimate 522 in base sales if we stock R.\nWe estimate 4504 in base sales if we stock S.\nWe estimate 3415 in base sales if we stock T.\nWe record A as weighing 6078 toward the shelf capacity.\nWe record B as weighing 6077 toward the shelf capacity.\nWe record C as weighing 2845 toward the shelf capacity.\nWe record D as weighing 5410 toward the shelf capacity.\nWe record E as weighing 1306 toward the shelf capacity.\nWe record F as weighing 2770 toward the shelf capacity.\nWe record G as weighing 1319 toward the shelf capacity.\nWe record H as weighing 5593 toward the shelf capacity.\nWe record I as weighing 2312 toward the shelf capacity.\nWe record J as weighing 3040 toward the shelf capacity.\nWe record K as weighing 2556 toward the shelf capacity.\nWe record L as weighing 3821 toward the shelf capacity.\nWe record M as weighing 669 toward the shelf capacity.\nWe record N as weighing 627 toward the shelf capacity.\nWe record O as weighing 6445 toward the shelf capacity.\nWe record P as weighing 3115 toward the shelf capacity.\nWe record Q as weighing 5814 toward the shelf capacity.\nWe record R as weighing 4255 toward the shelf capacity.\nWe record S as weighing 3673 toward the shelf capacity.\nWe record T as weighing 4596 toward the shelf capacity.\nWe estimate an extra 6018 in sales if we stock A together with B.\nWe estimate an extra 6078 in sales if we stock A together with D.\nWe estimate an extra 4895 in sales if we stock A together with I.\nWe estimate an extra 5441 in sales if we stock A together with N.\nWe estimate an extra 651 in sales if we stock B together with F.\nWe estimate an extra 1413 in sales if we stock B together with J.\nWe estimate an extra 2847 in sales if we stock B together with K.\nWe estimate an extra 5697 in sales if we stock B together with P.\nWe estimate an extra 1840 in sales if we stock B together with Q.\nWe estimate an extra 2803 in sales if we stock B together with R.\nWe estimate an extra 5714 in sales if we stock B together with T.\nWe estimate an extra 497 in sales if we stock C together with F.\nWe estimate an extra 2845 in sales if we stock C together with G.\nWe estimate an extra 1955 in sales if we stock C together with O.\nWe estimate an extra 2845 in sales if we stock C together with Q.\nWe estimate an extra 1216 in sales if we stock C together with R.\nWe estimate an extra 4818 in sales if we stock D together with K.\nWe estimate an extra 5410 in sales if we stock D together with Q.\nWe estimate an extra 1306 in sales if we stock E together with J.\nWe estimate an extra 1306 in sales if we stock E together with K.\nWe estimate an extra 1306 in sales if we stock E together with O.\nWe estimate an extra 1306 in sales if we stock E together with P.\nWe estimate an extra 2770 in sales if we stock F together with G.\nWe estimate an extra 2770 in sales if we stock F together with K.\nWe estimate an extra 1319 in sales if we stock G together with J.\nWe estimate an extra 216 in sales if we stock G together with M.\nWe estimate an extra 755 in sales if we stock G together with O.\nWe estimate an extra 1741 in sales if we stock H together with N.\nWe estimate an extra 2803 in sales if we stock H together with O.\nWe estimate an extra 1346 in sales if we stock H together with P.\nWe estimate an extra 3453 in sales if we stock H together with S.\nWe estimate an extra 2312 in sales if we stock I together with M.\nWe estimate an extra 3918 in sales if we stock I together with Q.\nWe estimate an extra 4338 in sales if we stock I together with R.\nWe estimate an extra 3040 in sales if we stock J together with K.\nWe estimate an extra 3040 in sales if we stock J together with L.\nWe estimate an extra 3040 in sales if we stock J together with N.\nWe estimate an extra 2026 in sales if we stock J together with O.\nWe estimate an extra 2556 in sales if we stock K together with O.\nWe estimate an extra 3519 in sales if we stock K together with Q.\nWe estimate an extra 3653 in sales if we stock L together with M.\nWe estimate an extra 3821 in sales if we stock L together with N.\nWe estimate an extra 124 in sales if we stock L together with Q.\nWe estimate an extra 5521 in sales if we stock M together with Q.\nWe estimate an extra 3574 in sales if we stock N together with Q.\nWe estimate an extra 1216 in sales if we stock R together with S.\nWe estimate an extra 3176 in sales if we stock S together with T.\nWe’ll use these details to choose which products to place so we maximize total sales including any pairwise bonuses.\n\nAlso, to keep things machine-friendly, please put your chosen items into a simple JSON object like this:\n\n{\n  \"solution\": [<selected_product_id>, <selected_product_id>, ...]\n}\n\nHere \"solution\" is just the list of product IDs you want to feature on the shelf — one entry per chosen product. Think of it like a short form: the array lists which products go on the shelf, and the order doesn't matter. This JSON is only a sketch of the shape I need, not the final answer itself.\n\nAll identifiers must be used exactly as they appear in the instance input — no renaming and no new labels.\n- for example: \"Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.\"",
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        5308,
        2582,
        4728,
        1526,
        1683,
        2971,
        522,
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      "weights": [
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      "capacity": 70207,
      "solution": [
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      "obj": 185987.0,
      "problem_type": "QKP"
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    "solution": [
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    "obj": 185987.0,
    "instance_variant": {
      "problem_type": "QKP",
      "num_items": 20,
      "capacity": 70207,
      "items": [
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        "B",
        "C",
        "D",
        "E",
        "F",
        "G",
        "H",
        "I",
        "J",
        "K",
        "L",
        "M",
        "N",
        "O",
        "P",
        "Q",
        "R",
        "S",
        "T"
      ],
      "linear_pairs": [
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          "item_id": "A",
          "linear_profit": 3913
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        {
          "item_id": "B",
          "linear_profit": 2255
        },
        {
          "item_id": "C",
          "linear_profit": 1503
        },
        {
          "item_id": "D",
          "linear_profit": 5766
        },
        {
          "item_id": "E",
          "linear_profit": 4188
        },
        {
          "item_id": "F",
          "linear_profit": 2000
        },
        {
          "item_id": "G",
          "linear_profit": 3309
        },
        {
          "item_id": "H",
          "linear_profit": 4455
        },
        {
          "item_id": "I",
          "linear_profit": 1277
        },
        {
          "item_id": "J",
          "linear_profit": 1142
        },
        {
          "item_id": "K",
          "linear_profit": 3374
        },
        {
          "item_id": "L",
          "linear_profit": 5308
        },
        {
          "item_id": "M",
          "linear_profit": 2582
        },
        {
          "item_id": "N",
          "linear_profit": 4728
        },
        {
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          "linear_profit": 1526
        },
        {
          "item_id": "P",
          "linear_profit": 1683
        },
        {
          "item_id": "Q",
          "linear_profit": 2971
        },
        {
          "item_id": "R",
          "linear_profit": 522
        },
        {
          "item_id": "S",
          "linear_profit": 4504
        },
        {
          "item_id": "T",
          "linear_profit": 3415
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      ],
      "weight_pairs": [
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          "item_id": "A",
          "weight": 6078
        },
        {
          "item_id": "B",
          "weight": 6077
        },
        {
          "item_id": "C",
          "weight": 2845
        },
        {
          "item_id": "D",
          "weight": 5410
        },
        {
          "item_id": "E",
          "weight": 1306
        },
        {
          "item_id": "F",
          "weight": 2770
        },
        {
          "item_id": "G",
          "weight": 1319
        },
        {
          "item_id": "H",
          "weight": 5593
        },
        {
          "item_id": "I",
          "weight": 2312
        },
        {
          "item_id": "J",
          "weight": 3040
        },
        {
          "item_id": "K",
          "weight": 2556
        },
        {
          "item_id": "L",
          "weight": 3821
        },
        {
          "item_id": "M",
          "weight": 669
        },
        {
          "item_id": "N",
          "weight": 627
        },
        {
          "item_id": "O",
          "weight": 6445
        },
        {
          "item_id": "P",
          "weight": 3115
        },
        {
          "item_id": "Q",
          "weight": 5814
        },
        {
          "item_id": "R",
          "weight": 4255
        },
        {
          "item_id": "S",
          "weight": 3673
        },
        {
          "item_id": "T",
          "weight": 4596
        }
      ],
      "quadratic_pairs": [
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          "item_i_id": "A",
          "item_j_id": "A",
          "quadratic_profit": 1619
        },
        {
          "item_i_id": "A",
          "item_j_id": "B",
          "quadratic_profit": 6018
        },
        {
          "item_i_id": "A",
          "item_j_id": "D",
          "quadratic_profit": 6078
        },
        {
          "item_i_id": "A",
          "item_j_id": "I",
          "quadratic_profit": 4895
        },
        {
          "item_i_id": "A",
          "item_j_id": "N",
          "quadratic_profit": 5441
        },
        {
          "item_i_id": "B",
          "item_j_id": "B",
          "quadratic_profit": 2845
        },
        {
          "item_i_id": "B",
          "item_j_id": "F",
          "quadratic_profit": 651
        },
        {
          "item_i_id": "B",
          "item_j_id": "J",
          "quadratic_profit": 1413
        },
        {
          "item_i_id": "B",
          "item_j_id": "K",
          "quadratic_profit": 2847
        },
        {
          "item_i_id": "B",
          "item_j_id": "P",
          "quadratic_profit": 5697
        },
        {
          "item_i_id": "B",
          "item_j_id": "Q",
          "quadratic_profit": 1840
        },
        {
          "item_i_id": "B",
          "item_j_id": "R",
          "quadratic_profit": 2803
        },
        {
          "item_i_id": "B",
          "item_j_id": "T",
          "quadratic_profit": 5714
        },
        {
          "item_i_id": "C",
          "item_j_id": "F",
          "quadratic_profit": 497
        },
        {
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          "item_j_id": "G",
          "quadratic_profit": 2845
        },
        {
          "item_i_id": "C",
          "item_j_id": "O",
          "quadratic_profit": 1955
        },
        {
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          "item_j_id": "Q",
          "quadratic_profit": 2845
        },
        {
          "item_i_id": "C",
          "item_j_id": "R",
          "quadratic_profit": 1216
        },
        {
          "item_i_id": "D",
          "item_j_id": "K",
          "quadratic_profit": 4818
        },
        {
          "item_i_id": "D",
          "item_j_id": "Q",
          "quadratic_profit": 5410
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        {
          "item_i_id": "E",
          "item_j_id": "E",
          "quadratic_profit": 1306
        },
        {
          "item_i_id": "E",
          "item_j_id": "J",
          "quadratic_profit": 1306
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        {
          "item_i_id": "E",
          "item_j_id": "K",
          "quadratic_profit": 1306
        },
        {
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          "item_j_id": "O",
          "quadratic_profit": 1306
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        {
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          "item_j_id": "P",
          "quadratic_profit": 1306
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        {
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          "item_j_id": "G",
          "quadratic_profit": 2770
        },
        {
          "item_i_id": "F",
          "item_j_id": "K",
          "quadratic_profit": 2770
        },
        {
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          "item_j_id": "J",
          "quadratic_profit": 1319
        },
        {
          "item_i_id": "G",
          "item_j_id": "M",
          "quadratic_profit": 216
        },
        {
          "item_i_id": "G",
          "item_j_id": "O",
          "quadratic_profit": 755
        },
        {
          "item_i_id": "H",
          "item_j_id": "N",
          "quadratic_profit": 1741
        },
        {
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          "item_j_id": "O",
          "quadratic_profit": 2803
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        {
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          "item_j_id": "P",
          "quadratic_profit": 1346
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        {
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          "item_j_id": "S",
          "quadratic_profit": 3453
        },
        {
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          "item_j_id": "I",
          "quadratic_profit": 931
        },
        {
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          "item_j_id": "M",
          "quadratic_profit": 2312
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        {
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          "item_j_id": "Q",
          "quadratic_profit": 3918
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        {
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          "item_j_id": "R",
          "quadratic_profit": 4338
        },
        {
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          "item_j_id": "J",
          "quadratic_profit": 315
        },
        {
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          "item_j_id": "K",
          "quadratic_profit": 3040
        },
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          "item_j_id": "L",
          "quadratic_profit": 3040
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          "item_j_id": "N",
          "quadratic_profit": 3040
        },
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          "item_i_id": "J",
          "item_j_id": "O",
          "quadratic_profit": 2026
        },
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          "item_j_id": "O",
          "quadratic_profit": 2556
        },
        {
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          "item_j_id": "Q",
          "quadratic_profit": 3519
        },
        {
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          "item_j_id": "M",
          "quadratic_profit": 3653
        },
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          "item_j_id": "N",
          "quadratic_profit": 3821
        },
        {
          "item_i_id": "L",
          "item_j_id": "Q",
          "quadratic_profit": 124
        },
        {
          "item_i_id": "M",
          "item_j_id": "M",
          "quadratic_profit": 669
        },
        {
          "item_i_id": "M",
          "item_j_id": "Q",
          "quadratic_profit": 5521
        },
        {
          "item_i_id": "N",
          "item_j_id": "N",
          "quadratic_profit": 375
        },
        {
          "item_i_id": "N",
          "item_j_id": "Q",
          "quadratic_profit": 3574
        },
        {
          "item_i_id": "R",
          "item_j_id": "S",
          "quadratic_profit": 1216
        },
        {
          "item_i_id": "S",
          "item_j_id": "T",
          "quadratic_profit": 3176
        }
      ]
    },
    "solution_variant": [
      "A",
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  {
    "task_name": "QKP",
    "problem_type": "QKP",
    "instruction": "On a tight schedule, the relief coordinator has one truck and several crates to pick from; each crate has its own weight and life‑saving worth, and a few specific pairs of crates give an extra lift to the relief effort if both are sent. The aim is straightforward in everyday terms: get the biggest total relief impact by adding each loaded crate’s value and any extra pair bonuses, while keeping the total load under the truck’s capacity — no crate can be taken more than once. The detailed list of crates, weights, values and pair bonuses follows below.\n\nBelow are the 21 crates (0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20) and the truck's weight capacity 741.\nCrate 0 has base life‑saving value 100.\nCrate 1 has base life‑saving value 103.\nCrate 2 has base life‑saving value 101.\nCrate 3 has base life‑saving value 104.\nCrate 4 has base life‑saving value 102.\nCrate 5 has base life‑saving value 103.\nCrate 6 has base life‑saving value 105.\nCrate 7 has base life‑saving value 104.\nCrate 8 has base life‑saving value 103.\nCrate 9 has base life‑saving value 103.\nCrate 10 has base life‑saving value 105.\nCrate 11 has base life‑saving value 101.\nCrate 12 has base life‑saving value 103.\nCrate 13 has base life‑saving value 105.\nCrate 14 has base life‑saving value 101.\nCrate 15 has base life‑saving value 100.\nCrate 16 has base life‑saving value 105.\nCrate 17 has base life‑saving value 102.\nCrate 18 has base life‑saving value 101.\nCrate 19 has base life‑saving value 101.\nCrate 20 has base life‑saving value 102.\nCrate 0 weighs 102.\nCrate 1 weighs 101.\nCrate 2 weighs 101.\nCrate 3 weighs 103.\nCrate 4 weighs 103.\nCrate 5 weighs 102.\nCrate 6 weighs 101.\nCrate 7 weighs 103.\nCrate 8 weighs 102.\nCrate 9 weighs 102.\nCrate 10 weighs 105.\nCrate 11 weighs 105.\nCrate 12 weighs 101.\nCrate 13 weighs 103.\nCrate 14 weighs 102.\nCrate 15 weighs 102.\nCrate 16 weighs 102.\nCrate 17 weighs 105.\nCrate 18 weighs 102.\nCrate 19 weighs 103.\nCrate 20 weighs 103.\nIf both crate 0 and crate 1 are loaded, they add 100 extra life‑saving value.\nIf both crate 0 and crate 2 are loaded, they add 103 extra life‑saving value.\nIf both crate 0 and crate 3 are loaded, they add 104 extra life‑saving value.\nIf both crate 0 and crate 5 are loaded, they add 102 extra life‑saving value.\nIf both crate 0 and crate 6 are loaded, they add 102 extra life‑saving value.\nIf both crate 0 and crate 7 are loaded, they add 100 extra life‑saving value.\nIf both crate 0 and crate 9 are loaded, they add 102 extra life‑saving value.\nIf both crate 0 and crate 10 are loaded, they add 105 extra life‑saving value.\nIf both crate 0 and crate 18 are loaded, they add 101 extra life‑saving value.\nIf both crate 0 and crate 19 are loaded, they add 104 extra life‑saving value.\nIf both crate 1 and crate 2 are loaded, they add 105 extra life‑saving value.\nIf both crate 1 and crate 3 are loaded, they add 105 extra life‑saving value.\nIf both crate 1 and crate 4 are loaded, they add 104 extra life‑saving value.\nIf both crate 1 and crate 6 are loaded, they add 105 extra life‑saving value.\nIf both crate 1 and crate 7 are loaded, they add 103 extra life‑saving value.\nIf both crate 1 and crate 8 are loaded, they add 100 extra life‑saving value.\nIf both crate 1 and crate 9 are loaded, they add 105 extra life‑saving value.\nIf both crate 1 and crate 13 are loaded, they add 102 extra life‑saving value.\nIf both crate 1 and crate 14 are loaded, they add 100 extra life‑saving value.\nIf both crate 1 and crate 15 are loaded, they add 104 extra life‑saving value.\nIf both crate 1 and crate 16 are loaded, they add 105 extra life‑saving value.\nIf both crate 1 and crate 17 are loaded, they add 105 extra life‑saving value.\nIf both crate 1 and crate 18 are loaded, they add 105 extra life‑saving value.\nIf both crate 2 and crate 3 are loaded, they add 100 extra life‑saving value.\nIf both crate 2 and crate 5 are loaded, they add 101 extra life‑saving value.\nIf both crate 2 and crate 6 are loaded, they add 102 extra life‑saving value.\nIf both crate 2 and crate 7 are loaded, they add 103 extra life‑saving value.\nIf both crate 2 and crate 8 are loaded, they add 105 extra life‑saving value.\nIf both crate 2 and crate 17 are loaded, they add 104 extra life‑saving value.\nIf both crate 2 and crate 18 are loaded, they add 105 extra life‑saving value.\nIf both crate 2 and crate 19 are loaded, they add 101 extra life‑saving value.\nIf both crate 2 and crate 20 are loaded, they add 101 extra life‑saving value.\nIf both crate 3 and crate 6 are loaded, they add 100 extra life‑saving value.\nIf both crate 3 and crate 9 are loaded, they add 100 extra life‑saving value.\nIf both crate 3 and crate 11 are loaded, they add 105 extra life‑saving value.\nIf both crate 3 and crate 13 are loaded, they add 100 extra life‑saving value.\nIf both crate 3 and crate 14 are loaded, they add 104 extra life‑saving value.\nIf both crate 3 and crate 15 are loaded, they add 101 extra life‑saving value.\nIf both crate 3 and crate 16 are loaded, they add 103 extra life‑saving value.\nIf both crate 3 and crate 19 are loaded, they add 105 extra life‑saving value.\nIf both crate 4 and crate 5 are loaded, they add 103 extra life‑saving value.\nIf both crate 4 and crate 7 are loaded, they add 103 extra life‑saving value.\nIf both crate 4 and crate 8 are loaded, they add 101 extra life‑saving value.\nIf both crate 4 and crate 10 are loaded, they add 101 extra life‑saving value.\nIf both crate 4 and crate 12 are loaded, they add 100 extra life‑saving value.\nIf both crate 4 and crate 13 are loaded, they add 104 extra life‑saving value.\nIf both crate 4 and crate 19 are loaded, they add 102 extra life‑saving value.\nIf both crate 5 and crate 7 are loaded, they add 104 extra life‑saving value.\nIf both crate 5 and crate 10 are loaded, they add 100 extra life‑saving value.\nIf both crate 5 and crate 11 are loaded, they add 103 extra life‑saving value.\nIf both crate 5 and crate 12 are loaded, they add 100 extra life‑saving value.\nIf both crate 5 and crate 13 are loaded, they add 105 extra life‑saving value.\nIf both crate 5 and crate 14 are loaded, they add 103 extra life‑saving value.\nIf both crate 5 and crate 16 are loaded, they add 101 extra life‑saving value.\nIf both crate 5 and crate 17 are loaded, they add 101 extra life‑saving value.\nIf both crate 5 and crate 18 are loaded, they add 105 extra life‑saving value.\nIf both crate 5 and crate 19 are loaded, they add 105 extra life‑saving value.\nIf both crate 5 and crate 20 are loaded, they add 104 extra life‑saving value.\nIf both crate 6 and crate 7 are loaded, they add 100 extra life‑saving value.\nIf both crate 6 and crate 10 are loaded, they add 104 extra life‑saving value.\nIf both crate 6 and crate 15 are loaded, they add 102 extra life‑saving value.\nIf both crate 6 and crate 17 are loaded, they add 105 extra life‑saving value.\nIf both crate 6 and crate 18 are loaded, they add 103 extra life‑saving value.\nIf both crate 6 and crate 20 are loaded, they add 101 extra life‑saving value.\nIf both crate 7 and crate 10 are loaded, they add 105 extra life‑saving value.\nIf both crate 7 and crate 11 are loaded, they add 101 extra life‑saving value.\nIf both crate 7 and crate 12 are loaded, they add 104 extra life‑saving value.\nIf both crate 7 and crate 15 are loaded, they add 101 extra life‑saving value.\nIf both crate 7 and crate 18 are loaded, they add 100 extra life‑saving value.\nIf both crate 8 and crate 9 are loaded, they add 100 extra life‑saving value.\nIf both crate 8 and crate 12 are loaded, they add 102 extra life‑saving value.\nIf both crate 8 and crate 14 are loaded, they add 103 extra life‑saving value.\nIf both crate 8 and crate 15 are loaded, they add 101 extra life‑saving value.\nIf both crate 8 and crate 17 are loaded, they add 101 extra life‑saving value.\nIf both crate 8 and crate 18 are loaded, they add 102 extra life‑saving value.\nIf both crate 8 and crate 20 are loaded, they add 105 extra life‑saving value.\nIf both crate 9 and crate 11 are loaded, they add 104 extra life‑saving value.\nIf both crate 9 and crate 13 are loaded, they add 105 extra life‑saving value.\nIf both crate 9 and crate 14 are loaded, they add 105 extra life‑saving value.\nIf both crate 9 and crate 15 are loaded, they add 101 extra life‑saving value.\nIf both crate 9 and crate 17 are loaded, they add 103 extra life‑saving value.\nIf both crate 9 and crate 18 are loaded, they add 102 extra life‑saving value.\nIf both crate 9 and crate 20 are loaded, they add 101 extra life‑saving value.\nIf both crate 10 and crate 11 are loaded, they add 103 extra life‑saving value.\nIf both crate 10 and crate 14 are loaded, they add 103 extra life‑saving value.\nIf both crate 10 and crate 15 are loaded, they add 101 extra life‑saving value.\nIf both crate 10 and crate 16 are loaded, they add 100 extra life‑saving value.\nIf both crate 10 and crate 17 are loaded, they add 102 extra life‑saving value.\nIf both crate 10 and crate 18 are loaded, they add 100 extra life‑saving value.\nIf both crate 11 and crate 13 are loaded, they add 102 extra life‑saving value.\nIf both crate 11 and crate 14 are loaded, they add 104 extra life‑saving value.\nIf both crate 11 and crate 15 are loaded, they add 102 extra life‑saving value.\nIf both crate 11 and crate 16 are loaded, they add 101 extra life‑saving value.\nIf both crate 11 and crate 17 are loaded, they add 101 extra life‑saving value.\nIf both crate 11 and crate 18 are loaded, they add 104 extra life‑saving value.\nIf both crate 12 and crate 15 are loaded, they add 100 extra life‑saving value.\nIf both crate 12 and crate 17 are loaded, they add 103 extra life‑saving value.\nIf both crate 12 and crate 18 are loaded, they add 100 extra life‑saving value.\nIf both crate 12 and crate 19 are loaded, they add 100 extra life‑saving value.\nIf both crate 12 and crate 20 are loaded, they add 103 extra life‑saving value.\nIf both crate 13 and crate 14 are loaded, they add 105 extra life‑saving value.\nIf both crate 13 and crate 15 are loaded, they add 103 extra life‑saving value.\nIf both crate 13 and crate 17 are loaded, they add 102 extra life‑saving value.\nIf both crate 13 and crate 18 are loaded, they add 102 extra life‑saving value.\nIf both crate 14 and crate 20 are loaded, they add 100 extra life‑saving value.\nIf both crate 15 and crate 16 are loaded, they add 101 extra life‑saving value.\nIf both crate 15 and crate 18 are loaded, they add 104 extra life‑saving value.\nIf both crate 16 and crate 17 are loaded, they add 103 extra life‑saving value.\nIf both crate 16 and crate 18 are loaded, they add 100 extra life‑saving value.\nIf both crate 17 and crate 20 are loaded, they add 101 extra life‑saving value.\nIf both crate 18 and crate 19 are loaded, they add 101 extra life‑saving value.\nIf both crate 19 and crate 20 are loaded, they add 105 extra life‑saving value.\nThe relief coordinator will select crates so total weight stays within 741 while maximizing life‑saving impact.\n\nIf you want to hand me the picks in a machine-friendly way, a little JSON snippet like this works nicely — just list the crate IDs you plan to load.\n\n{\n  \"solution\": [<selected_crate_id>, <selected_crate_id>, ...]\n}\n\nThink of it like a simple form: \"solution\" is the list of crates you choose to put on the truck, and each angle-bracket entry is where you drop the exact crate identifier from the instance. This JSON is just a sketch of the shape I expect — not the actual chosen crates.\n\nPlease be careful to use the identifiers exactly as they appear in the instance input — no renaming and no new labels.\n- for example: \"Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.\"",
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      "num_items": 21,
      "capacity": 741,
      "items": [
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  {
    "task_name": "QKP",
    "problem_type": "QKP",
    "instruction": "At the community theater, the stage manager is figuring out which flats, props and backdrops to load into the single available truck. Every item uses up some of the truck’s capacity and brings a rental value, and a few pairings create extra payoff when both pieces arrive because they enhance the scene together. The sensible plan is to pick items that don’t overload the truck and that make the most total return — that total being each chosen piece’s rental added together plus any extra payoffs from pairs that are both taken. Each piece is taken once or left behind; the exact item weights, rental values and pair bonuses are listed below.\n\n{\n  \"total_pieces\": 19,\n  \"truck_load_limit\": 3324,\n  \"piece_ids\": [\n    1,\n    2,\n    3,\n    4,\n    5,\n    6,\n    7,\n    8,\n    9,\n    10,\n    11,\n    12,\n    13,\n    14,\n    15,\n    16,\n    17,\n    18,\n    19\n  ],\n  \"linear\": [\n    {\n      \"piece_id\": 1,\n      \"rental_value\": 179\n    },\n    {\n      \"piece_id\": 2,\n      \"rental_value\": 152\n    },\n    {\n      \"piece_id\": 3,\n      \"rental_value\": 233\n    },\n    {\n      \"piece_id\": 4,\n      \"rental_value\": 270\n    },\n    {\n      \"piece_id\": 5,\n      \"rental_value\": 274\n    },\n    {\n      \"piece_id\": 6,\n      \"rental_value\": 241\n    },\n    {\n      \"piece_id\": 7,\n      \"rental_value\": 224\n    },\n    {\n      \"piece_id\": 8,\n      \"rental_value\": 250\n    },\n    {\n      \"piece_id\": 9,\n      \"rental_value\": 297\n    },\n    {\n      \"piece_id\": 10,\n      \"rental_value\": 322\n    },\n    {\n      \"piece_id\": 11,\n      \"rental_value\": 199\n    },\n    {\n      \"piece_id\": 12,\n      \"rental_value\": 289\n    },\n    {\n      \"piece_id\": 13,\n      \"rental_value\": 234\n    },\n    {\n      \"piece_id\": 14,\n      \"rental_value\": 219\n    },\n    {\n      \"piece_id\": 15,\n      \"rental_value\": 248\n    },\n    {\n      \"piece_id\": 16,\n      \"rental_value\": 143\n    },\n    {\n      \"piece_id\": 17,\n      \"rental_value\": 140\n    },\n    {\n      \"piece_id\": 18,\n      \"rental_value\": 270\n    },\n    {\n      \"piece_id\": 19,\n      \"rental_value\": 285\n    }\n  ],\n  \"weights\": [\n    {\n      \"piece_id\": 1,\n      \"piece_weight\": 149\n    },\n    {\n      \"piece_id\": 2,\n      \"piece_weight\": 122\n    },\n    {\n      \"piece_id\": 3,\n      \"piece_weight\": 203\n    },\n    {\n      \"piece_id\": 4,\n      \"piece_weight\": 240\n    },\n    {\n      \"piece_id\": 5,\n      \"piece_weight\": 244\n    },\n    {\n      \"piece_id\": 6,\n      \"piece_weight\": 211\n    },\n    {\n      \"piece_id\": 7,\n      \"piece_weight\": 194\n    },\n    {\n      \"piece_id\": 8,\n      \"piece_weight\": 220\n    },\n    {\n      \"piece_id\": 9,\n      \"piece_weight\": 267\n    },\n    {\n      \"piece_id\": 10,\n      \"piece_weight\": 292\n    },\n    {\n      \"piece_id\": 11,\n      \"piece_weight\": 169\n    },\n    {\n      \"piece_id\": 12,\n      \"piece_weight\": 259\n    },\n    {\n      \"piece_id\": 13,\n      \"piece_weight\": 204\n    },\n    {\n      \"piece_id\": 14,\n      \"piece_weight\": 189\n    },\n    {\n      \"piece_id\": 15,\n      \"piece_weight\": 218\n    },\n    {\n      \"piece_id\": 16,\n      \"piece_weight\": 113\n    },\n    {\n      \"piece_id\": 17,\n      \"piece_weight\": 110\n    },\n    {\n      \"piece_id\": 18,\n      \"piece_weight\": 240\n    },\n    {\n      \"piece_id\": 19,\n      \"piece_weight\": 255\n    }\n  ],\n  \"quadratic\": [\n    {\n      \"piece_i_id\": 1,\n      \"piece_j_id\": 5,\n      \"pair_synergy_value\": 169\n    },\n    {\n      \"piece_i_id\": 1,\n      \"piece_j_id\": 7,\n      \"pair_synergy_value\": 138\n    },\n    {\n      \"piece_i_id\": 1,\n      \"piece_j_id\": 8,\n      \"pair_synergy_value\": 243\n    },\n    {\n      \"piece_i_id\": 1,\n      \"piece_j_id\": 9,\n      \"pair_synergy_value\": 274\n    },\n    {\n      \"piece_i_id\": 1,\n      \"piece_j_id\": 10,\n      \"pair_synergy_value\": 236\n    },\n    {\n      \"piece_i_id\": 1,\n      \"piece_j_id\": 13,\n      \"pair_synergy_value\": 138\n    },\n    {\n      \"piece_i_id\": 1,\n      \"piece_j_id\": 15,\n      \"pair_synergy_value\": 190\n    },\n    {\n      \"piece_i_id\": 1,\n      \"piece_j_id\": 16,\n      \"pair_synergy_value\": 291\n    },\n    {\n      \"piece_i_id\": 1,\n      \"piece_j_id\": 17,\n      \"pair_synergy_value\": 185\n    },\n    {\n      \"piece_i_id\": 2,\n      \"piece_j_id\": 3,\n      \"pair_synergy_value\": 154\n    },\n    {\n      \"piece_i_id\": 2,\n      \"piece_j_id\": 4,\n      \"pair_synergy_value\": 193\n    },\n    {\n      \"piece_i_id\": 2,\n      \"piece_j_id\": 6,\n      \"pair_synergy_value\": 263\n    },\n    {\n      \"piece_i_id\": 2,\n      \"piece_j_id\": 7,\n      \"pair_synergy_value\": 262\n    },\n    {\n      \"piece_i_id\": 2,\n      \"piece_j_id\": 8,\n      \"pair_synergy_value\": 176\n    },\n    {\n      \"piece_i_id\": 2,\n      \"piece_j_id\": 14,\n      \"pair_synergy_value\": 287\n    },\n    {\n      \"piece_i_id\": 2,\n      \"piece_j_id\": 15,\n      \"pair_synergy_value\": 236\n    },\n    {\n      \"piece_i_id\": 2,\n      \"piece_j_id\": 17,\n      \"pair_synergy_value\": 257\n    },\n    {\n      \"piece_i_id\": 3,\n      \"piece_j_id\": 8,\n      \"pair_synergy_value\": 221\n    },\n    {\n      \"piece_i_id\": 3,\n      \"piece_j_id\": 9,\n      \"pair_synergy_value\": 122\n    },\n    {\n      \"piece_i_id\": 3,\n      \"piece_j_id\": 10,\n      \"pair_synergy_value\": 159\n    },\n    {\n      \"piece_i_id\": 3,\n      \"piece_j_id\": 11,\n      \"pair_synergy_value\": 137\n    },\n    {\n      \"piece_i_id\": 3,\n      \"piece_j_id\": 14,\n      \"pair_synergy_value\": 289\n    },\n    {\n      \"piece_i_id\": 3,\n      \"piece_j_id\": 15,\n      \"pair_synergy_value\": 260\n    },\n    {\n      \"piece_i_id\": 3,\n      \"piece_j_id\": 16,\n      \"pair_synergy_value\": 208\n    },\n    {\n      \"piece_i_id\": 3,\n      \"piece_j_id\": 19,\n      \"pair_synergy_value\": 139\n    },\n    {\n      \"piece_i_id\": 4,\n      \"piece_j_id\": 5,\n      \"pair_synergy_value\": 242\n    },\n    {\n      \"piece_i_id\": 4,\n      \"piece_j_id\": 10,\n      \"pair_synergy_value\": 129\n    },\n    {\n      \"piece_i_id\": 4,\n      \"piece_j_id\": 11,\n      \"pair_synergy_value\": 199\n    },\n    {\n      \"piece_i_id\": 4,\n      \"piece_j_id\": 12,\n  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\"pair_synergy_value\": 270\n    },\n    {\n      \"piece_i_id\": 9,\n      \"piece_j_id\": 10,\n      \"pair_synergy_value\": 282\n    },\n    {\n      \"piece_i_id\": 9,\n      \"piece_j_id\": 11,\n      \"pair_synergy_value\": 133\n    },\n    {\n      \"piece_i_id\": 9,\n      \"piece_j_id\": 13,\n      \"pair_synergy_value\": 264\n    },\n    {\n      \"piece_i_id\": 9,\n      \"piece_j_id\": 14,\n      \"pair_synergy_value\": 222\n    },\n    {\n      \"piece_i_id\": 9,\n      \"piece_j_id\": 15,\n      \"pair_synergy_value\": 276\n    },\n    {\n      \"piece_i_id\": 9,\n      \"piece_j_id\": 17,\n      \"pair_synergy_value\": 288\n    },\n    {\n      \"piece_i_id\": 10,\n      \"piece_j_id\": 12,\n      \"pair_synergy_value\": 261\n    },\n    {\n      \"piece_i_id\": 10,\n      \"piece_j_id\": 16,\n      \"pair_synergy_value\": 100\n    },\n    {\n      \"piece_i_id\": 10,\n      \"piece_j_id\": 17,\n      \"pair_synergy_value\": 261\n    },\n    {\n      \"piece_i_id\": 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        }
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    "solution_variant": [
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    "context_index": 38,
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  {
    "task_name": "QKP",
    "problem_type": "QKP",
    "instruction": "We’re curating a weekend gallery and can only ship what fits in one crate. Every artwork comes with an estimated sale value and takes up some space, and certain combinations of paintings boost interest when they’re displayed together, giving extra value. The idea is to pick which pieces to include so the crate stays within its capacity, no piece is repeated, and the total expected return (individual values plus any extra for specific pairs) is as large as it can be. The concrete weights, values, and synergy numbers are shown below.\n\n{\n  \"total_artworks\": 22,\n  \"crate_capacity\": 1258,\n  \"artwork_ids\": [\n    \"A\",\n    \"B\",\n    \"C\",\n    \"D\",\n    \"E\",\n    \"F\",\n    \"G\",\n    \"H\",\n    \"I\",\n    \"J\",\n    \"K\",\n    \"L\",\n    \"M\",\n    \"N\",\n    \"O\",\n    \"P\",\n    \"Q\",\n    \"R\",\n    \"S\",\n    \"T\",\n    \"U\",\n    \"V\"\n  ],\n  \"linear\": [\n    {\n      \"artwork_id\": \"A\",\n      \"expected_sale_value\": 104\n    },\n    {\n      \"artwork_id\": \"B\",\n      \"expected_sale_value\": 103\n    },\n    {\n      \"artwork_id\": \"C\",\n      \"expected_sale_value\": 101\n    },\n    {\n      \"artwork_id\": \"D\",\n      \"expected_sale_value\": 102\n    },\n    {\n      \"artwork_id\": \"E\",\n      \"expected_sale_value\": 103\n    },\n    {\n      \"artwork_id\": \"F\",\n      \"expected_sale_value\": 104\n    },\n    {\n      \"artwork_id\": \"G\",\n      \"expected_sale_value\": 101\n    },\n    {\n      \"artwork_id\": \"H\",\n      \"expected_sale_value\": 105\n    },\n    {\n      \"artwork_id\": \"I\",\n      \"expected_sale_value\": 101\n    },\n    {\n      \"artwork_id\": \"J\",\n      \"expected_sale_value\": 101\n    },\n    {\n      \"artwork_id\": \"K\",\n      \"expected_sale_value\": 103\n    },\n    {\n      \"artwork_id\": \"L\",\n      \"expected_sale_value\": 105\n    },\n    {\n      \"artwork_id\": \"M\",\n      \"expected_sale_value\": 104\n    },\n    {\n      \"artwork_id\": \"N\",\n      \"expected_sale_value\": 102\n    },\n    {\n      \"artwork_id\": \"O\",\n      \"expected_sale_value\": 103\n    },\n    {\n      \"artwork_id\": \"P\",\n      \"expected_sale_value\": 102\n    },\n    {\n      \"artwork_id\": \"Q\",\n      \"expected_sale_value\": 103\n    },\n    {\n      \"artwork_id\": \"R\",\n      \"expected_sale_value\": 100\n    },\n    {\n      \"artwork_id\": \"S\",\n      \"expected_sale_value\": 104\n    },\n    {\n      \"artwork_id\": \"T\",\n      \"expected_sale_value\": 103\n    },\n    {\n      \"artwork_id\": \"U\",\n      \"expected_sale_value\": 101\n    },\n    {\n      \"artwork_id\": \"V\",\n      \"expected_sale_value\": 105\n    }\n  ],\n  \"weights\": [\n    {\n      \"artwork_id\": \"A\",\n      \"space_required\": 102\n    },\n    {\n      \"artwork_id\": \"B\",\n      \"space_required\": 103\n    },\n    {\n      \"artwork_id\": \"C\",\n      \"space_required\": 101\n    },\n    {\n      \"artwork_id\": \"D\",\n      \"space_required\": 103\n    },\n    {\n      \"artwork_id\": \"E\",\n      \"space_required\": 105\n    },\n    {\n      \"artwork_id\": \"F\",\n      \"space_required\": 105\n    },\n    {\n      \"artwork_id\": \"G\",\n      \"space_required\": 103\n    },\n    {\n      \"artwork_id\": \"H\",\n      \"space_required\": 104\n    },\n    {\n      \"artwork_id\": \"I\",\n      \"space_required\": 104\n    },\n    {\n      \"artwork_id\": \"J\",\n      \"space_required\": 103\n    },\n    {\n      \"artwork_id\": \"K\",\n      \"space_required\": 100\n    },\n    {\n      \"artwork_id\": \"L\",\n      \"space_required\": 103\n    },\n    {\n      \"artwork_id\": \"M\",\n      \"space_required\": 101\n    },\n    {\n      \"artwork_id\": \"N\",\n      \"space_required\": 102\n    },\n    {\n      \"artwork_id\": \"O\",\n      \"space_required\": 105\n    },\n    {\n      \"artwork_id\": \"P\",\n      \"space_required\": 101\n    },\n    {\n      \"artwork_id\": \"Q\",\n      \"space_required\": 102\n    },\n    {\n      \"artwork_id\": \"R\",\n      \"space_required\": 101\n    },\n    {\n      \"artwork_id\": \"S\",\n      \"space_required\": 101\n    },\n    {\n      \"artwork_id\": \"T\",\n      \"space_required\": 102\n    },\n    {\n      \"artwork_id\": \"U\",\n      \"space_required\": 105\n    },\n    {\n      \"artwork_id\": \"V\",\n      \"space_required\": 100\n    }\n  ],\n  \"quadratic\": [\n    {\n      \"artwork_i_id\": \"A\",\n      \"artwork_j_id\": \"B\",\n      \"synergy_bonus_value\": 102\n    },\n    {\n      \"artwork_i_id\": \"A\",\n      \"artwork_j_id\": \"C\",\n      \"synergy_bonus_value\": 100\n    },\n    {\n      \"artwork_i_id\": \"A\",\n      \"artwork_j_id\": \"F\",\n      \"synergy_bonus_value\": 101\n    },\n    {\n      \"artwork_i_id\": \"A\",\n      \"artwork_j_id\": \"G\",\n      \"synergy_bonus_value\": 102\n    },\n    {\n      \"artwork_i_id\": \"A\",\n      \"artwork_j_id\": \"H\",\n      \"synergy_bonus_value\": 102\n    },\n    {\n      \"artwork_i_id\": \"A\",\n      \"artwork_j_id\": \"I\",\n      \"synergy_bonus_value\": 102\n    },\n    {\n      \"artwork_i_id\": \"A\",\n      \"artwork_j_id\": \"K\",\n      \"synergy_bonus_value\": 100\n    },\n    {\n      \"artwork_i_id\": \"A\",\n      \"artwork_j_id\": \"L\",\n      \"synergy_bonus_value\": 103\n    },\n    {\n      \"artwork_i_id\": \"A\",\n      \"artwork_j_id\": \"M\",\n      \"synergy_bonus_value\": 101\n    },\n    {\n      \"artwork_i_id\": \"A\",\n      \"artwork_j_id\": \"N\",\n      \"synergy_bonus_value\": 102\n    },\n    {\n      \"artwork_i_id\": \"A\",\n      \"artwork_j_id\": \"O\",\n      \"synergy_bonus_value\": 102\n    },\n    {\n      \"artwork_i_id\": \"A\",\n      \"artwork_j_id\": \"P\",\n      \"synergy_bonus_value\": 102\n    },\n    {\n      \"artwork_i_id\": \"A\",\n      \"artwork_j_id\": \"Q\",\n      \"synergy_bonus_value\": 102\n    },\n    {\n      \"artwork_i_id\": \"A\",\n      \"artwork_j_id\": \"S\",\n      \"synergy_bonus_value\": 102\n    },\n    {\n      \"artwork_i_id\": \"A\",\n      \"artwork_j_id\": \"T\",\n      \"synergy_bonus_value\": 100\n    },\n    {\n      \"artwork_i_id\": \"A\",\n      \"artwork_j_id\": \"V\",\n      \"synergy_bonus_value\": 100\n    },\n    {\n      \"artwork_i_id\": \"B\",\n      \"artwork_j_id\": \"D\",\n      \"synergy_bonus_value\": 103\n    },\n    {\n      \"artwork_i_id\": \"B\",\n      \"artwork_j_id\": \"F\",\n      \"synergy_bonus_value\": 103\n    },\n    {\n      \"artwork_i_id\": \"B\",\n      \"artwork_j_id\": \"L\",\n      \"synergy_bonus_value\": 101\n    },\n    {\n      \"artwork_i_id\": \"B\",\n      \"artwork_j_id\": \"N\",\n      \"synergy_bonus_value\": 101\n    },\n    {\n      \"artwork_i_id\": \"B\",\n      \"artwork_j_id\": \"Q\",\n      \"synergy_bonus_value\": 101\n    },\n    {\n      \"artwork_i_id\": \"B\",\n      \"artwork_j_id\": \"R\",\n      \"synergy_bonus_value\": 103\n    },\n    {\n      \"artwork_i_id\": \"B\",\n      \"artwork_j_id\": \"T\",\n      \"synergy_bonus_value\": 103\n    },\n    {\n      \"artwork_i_id\": \"B\",\n      \"artwork_j_id\": \"V\",\n      \"synergy_bonus_value\": 103\n    },\n    {\n      \"artwork_i_id\": \"C\",\n      \"artwork_j_id\": \"D\",\n      \"synergy_bonus_value\": 101\n    },\n    {\n      \"artwork_i_id\": \"C\",\n      \"artwork_j_id\": \"F\",\n      \"synergy_bonus_value\": 101\n    },\n    {\n      \"artwork_i_id\": \"C\",\n      \"artwork_j_id\": \"I\",\n      \"synergy_bonus_value\": 101\n    },\n    {\n      \"artwork_i_id\": \"C\",\n      \"artwork_j_id\": \"P\",\n      \"synergy_bonus_value\": 100\n    },\n    {\n      \"artwork_i_id\": \"C\",\n      \"artwork_j_id\": \"Q\",\n      \"synergy_bonus_value\": 101\n    },\n    {\n      \"artwork_i_id\": \"C\",\n      \"artwork_j_id\": \"R\",\n      \"synergy_bonus_value\": 101\n    },\n    {\n      \"artwork_i_id\": \"C\",\n      \"artwork_j_id\": \"S\",\n      \"synergy_bonus_value\": 103\n    },\n    {\n      \"artwork_i_id\": \"C\",\n      \"artwork_j_id\": \"T\",\n      \"synergy_bonus_value\": 101\n    },\n    {\n      \"artwork_i_id\": \"C\",\n      \"artwork_j_id\": \"V\",\n      \"synergy_bonus_value\": 103\n    },\n    {\n      \"artwork_i_id\": \"D\",\n      \"artwork_j_id\": \"F\",\n      \"synergy_bonus_value\": 101\n    },\n    {\n      \"artwork_i_id\": \"D\",\n      \"artwork_j_id\": \"H\",\n      \"synergy_bonus_value\": 103\n    },\n    {\n      \"artwork_i_id\": \"D\",\n      \"artwork_j_id\": \"L\",\n      \"synergy_bonus_value\": 103\n    },\n    {\n      \"artwork_i_id\": \"D\",\n      \"artwork_j_id\": \"O\",\n      \"synergy_bonus_value\": 103\n    },\n    {\n      \"artwork_i_id\": \"D\",\n      \"artwork_j_id\": \"T\",\n      \"synergy_bonus_value\": 102\n    },\n    {\n      \"artwork_i_id\": \"D\",\n      \"artwork_j_id\": \"U\",\n      \"synergy_bonus_value\": 101\n    },\n    {\n      \"artwork_i_id\": \"E\",\n      \"artwork_j_id\": \"F\",\n      \"synergy_bonus_value\": 104\n    },\n    {\n      \"artwork_i_id\": \"E\",\n      \"artwork_j_id\": \"G\",\n      \"synergy_bonus_value\": 103\n    },\n    {\n      \"artwork_i_id\": \"E\",\n      \"artwork_j_id\": \"N\",\n      \"synergy_bonus_value\": 103\n    },\n    {\n      \"artwork_i_id\": \"E\",\n      \"artwork_j_id\": \"R\",\n      \"synergy_bonus_value\": 101\n    },\n    {\n      \"artwork_i_id\": \"E\",\n      \"artwork_j_id\": \"T\",\n      \"synergy_bonus_value\": 102\n    },\n    {\n      \"artwork_i_id\": \"E\",\n      \"artwork_j_id\": \"V\",\n      \"synergy_bonus_value\": 102\n    },\n    {\n      \"artwork_i_id\": \"F\",\n      \"artwork_j_id\": \"I\",\n      \"synergy_bonus_value\": 105\n    },\n    {\n      \"artwork_i_id\": \"F\",\n      \"artwork_j_id\": \"L\",\n      \"synergy_bonus_value\": 103\n    },\n    {\n      \"artwork_i_id\": \"F\",\n      \"artwork_j_id\": \"M\",\n      \"synergy_bonus_value\": 102\n    },\n    {\n      \"artwork_i_id\": \"F\",\n      \"artwork_j_id\": \"N\",\n      \"synergy_bonus_value\": 105\n    },\n    {\n      \"artwork_i_id\": \"F\",\n      \"artwork_j_id\": \"O\",\n      \"synergy_bonus_value\": 105\n    },\n    {\n      \"artwork_i_id\": \"F\",\n      \"artwork_j_id\": \"P\",\n      \"synergy_bonus_value\": 103\n    },\n    {\n      \"artwork_i_id\": \"F\",\n      \"artwork_j_id\": \"Q\",\n      \"synergy_bonus_value\": 100\n    },\n    {\n      \"artwork_i_id\": \"F\",\n      \"artwork_j_id\": \"R\",\n      \"synergy_bonus_value\": 101\n    },\n    {\n      \"artwork_i_id\": \"F\",\n      \"artwork_j_id\": \"S\",\n      \"synergy_bonus_value\": 102\n    },\n    {\n      \"artwork_i_id\": \"F\",\n      \"artwork_j_id\": \"U\",\n      \"synergy_bonus_value\": 100\n    },\n    {\n      \"artwork_i_id\": \"F\",\n      \"artwork_j_id\": \"V\",\n      \"synergy_bonus_value\": 101\n    },\n    {\n      \"artwork_i_id\": \"G\",\n      \"artwork_j_id\": \"H\",\n      \"synergy_bonus_value\": 101\n    },\n    {\n      \"artwork_i_id\": \"G\",\n      \"artwork_j_id\": \"J\",\n      \"synergy_bonus_value\": 101\n    },\n    {\n      \"artwork_i_id\": \"G\",\n      \"artwork_j_id\": \"P\",\n      \"synergy_bonus_value\": 102\n    },\n    {\n      \"artwork_i_id\": \"G\",\n      \"artwork_j_id\": \"R\",\n      \"synergy_bonus_value\": 101\n    },\n    {\n      \"artwork_i_id\": \"G\",\n      \"artwork_j_id\": \"S\",\n      \"synergy_bonus_value\": 101\n    },\n    {\n      \"artwork_i_id\": \"G\",\n      \"artwork_j_id\": \"T\",\n      \"synergy_bonus_value\": 101\n    },\n    {\n      \"artwork_i_id\": \"G\",\n      \"artwork_j_id\": \"V\",\n      \"synergy_bonus_value\": 102\n    },\n    {\n      \"artwork_i_id\": \"H\",\n      \"artwork_j_id\": \"M\",\n      \"synergy_bonus_value\": 102\n    },\n    {\n      \"artwork_i_id\": \"H\",\n      \"artwork_j_id\": \"N\",\n      \"synergy_bonus_value\": 102\n    },\n    {\n      \"artwork_i_id\": \"H\",\n      \"artwork_j_id\": \"O\",\n      \"synergy_bonus_value\": 104\n    },\n    {\n      \"artwork_i_id\": \"H\",\n      \"artwork_j_id\": \"Q\",\n      \"synergy_bonus_value\": 104\n    },\n    {\n      \"artwork_i_id\": \"H\",\n      \"artwork_j_id\": \"S\",\n      \"synergy_bonus_value\": 102\n    },\n    {\n      \"artwork_i_id\": \"H\",\n      \"artwork_j_id\": \"T\",\n      \"synergy_bonus_value\": 102\n    },\n    {\n      \"artwork_i_id\": \"H\",\n      \"artwork_j_id\": \"U\",\n      \"synergy_bonus_value\": 102\n    },\n    {\n      \"artwork_i_id\": \"I\",\n      \"artwork_j_id\": \"J\",\n      \"synergy_bonus_value\": 101\n    },\n    {\n      \"artwork_i_id\": \"I\",\n      \"artwork_j_id\": \"M\",\n      \"synergy_bonus_value\": 103\n    },\n    {\n      \"artwork_i_id\": \"I\",\n      \"artwork_j_id\": \"N\",\n      \"synergy_bonus_value\": 103\n    },\n    {\n      \"artwork_i_id\": \"I\",\n      \"artwork_j_id\": \"O\",\n      \"synergy_bonus_value\": 103\n    },\n    {\n      \"artwork_i_id\": \"I\",\n      \"artwork_j_id\": \"R\",\n      \"synergy_bonus_value\": 102\n    },\n    {\n      \"artwork_i_id\": \"I\",\n      \"artwork_j_id\": \"S\",\n      \"synergy_bonus_value\": 103\n    },\n    {\n      \"artwork_i_id\": \"I\",\n      \"artwork_j_id\": \"V\",\n      \"synergy_bonus_value\": 103\n    },\n    {\n      \"artwork_i_id\": \"J\",\n      \"artwork_j_id\": \"K\",\n      \"synergy_bonus_value\": 103\n    },\n    {\n      \"artwork_i_id\": \"J\",\n      \"artwork_j_id\": \"P\",\n      \"synergy_bonus_value\": 102\n    },\n    {\n      \"artwork_i_id\": \"J\",\n      \"artwork_j_id\": \"S\",\n      \"synergy_bonus_value\": 102\n    },\n    {\n      \"artwork_i_id\": \"J\",\n      \"artwork_j_id\": \"T\",\n      \"synergy_bonus_value\": 103\n    },\n    {\n      \"artwork_i_id\": \"J\",\n      \"artwork_j_id\": \"U\",\n      \"synergy_bonus_value\": 103\n    },\n    {\n      \"artwork_i_id\": \"K\",\n      \"artwork_j_id\": \"L\",\n      \"synergy_bonus_value\": 103\n    },\n    {\n      \"artwork_i_id\": \"K\",\n      \"artwork_j_id\": \"M\",\n      \"synergy_bonus_value\": 103\n    },\n    {\n      \"artwork_i_id\": \"K\",\n      \"artwork_j_id\": \"N\",\n      \"synergy_bonus_value\": 103\n    },\n    {\n      \"artwork_i_id\": \"K\",\n      \"artwork_j_id\": \"O\",\n      \"synergy_bonus_value\": 101\n    },\n    {\n      \"artwork_i_id\": \"K\",\n      \"artwork_j_id\": \"R\",\n      \"synergy_bonus_value\": 102\n    },\n    {\n  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\"synergy_bonus_value\": 101\n    },\n    {\n      \"artwork_i_id\": \"M\",\n      \"artwork_j_id\": \"Q\",\n      \"synergy_bonus_value\": 101\n    },\n    {\n      \"artwork_i_id\": \"M\",\n      \"artwork_j_id\": \"T\",\n      \"synergy_bonus_value\": 102\n    },\n    {\n      \"artwork_i_id\": \"N\",\n      \"artwork_j_id\": \"O\",\n      \"synergy_bonus_value\": 103\n    },\n    {\n      \"artwork_i_id\": \"N\",\n      \"artwork_j_id\": \"U\",\n      \"synergy_bonus_value\": 101\n    },\n    {\n      \"artwork_i_id\": \"N\",\n      \"artwork_j_id\": \"V\",\n      \"synergy_bonus_value\": 101\n    },\n    {\n      \"artwork_i_id\": \"O\",\n      \"artwork_j_id\": \"R\",\n      \"synergy_bonus_value\": 102\n    },\n    {\n      \"artwork_i_id\": \"P\",\n      \"artwork_j_id\": \"R\",\n      \"synergy_bonus_value\": 102\n    },\n    {\n      \"artwork_i_id\": \"P\",\n      \"artwork_j_id\": \"S\",\n      \"synergy_bonus_value\": 102\n    },\n    {\n      \"artwork_i_id\": \"P\",\n      \"artwork_j_id\": \"T\",\n      \"synergy_bonus_value\": 102\n    },\n    {\n      \"artwork_i_id\": \"P\",\n      \"artwork_j_id\": \"U\",\n      \"synergy_bonus_value\": 102\n    },\n    {\n      \"artwork_i_id\": \"P\",\n      \"artwork_j_id\": \"V\",\n      \"synergy_bonus_value\": 104\n    },\n    {\n      \"artwork_i_id\": \"Q\",\n      \"artwork_j_id\": \"S\",\n      \"synergy_bonus_value\": 102\n    },\n    {\n      \"artwork_i_id\": \"Q\",\n      \"artwork_j_id\": \"T\",\n      \"synergy_bonus_value\": 101\n    },\n    {\n      \"artwork_i_id\": \"Q\",\n      \"artwork_j_id\": \"V\",\n      \"synergy_bonus_value\": 102\n    }\n  ]\n}\n\nIf you want to give me your final pick, toss it in a tiny JSON snippet like this so it's easy to read:\n\n{\n  \"solution\": [<selected_artwork_id>, <selected_artwork_id>, ...]\n}\n\n\"solution\" is just a list of the artworks you're packing into the crate — each entry should be the identifier for a piece from the instance. Think of this JSON as a little form: the array holds the chosen items, and that's it. It's only a sketch of the shape I expect, not the actual answer.\n\nPlease make sure to use the exact identifiers from the instance input — don't rename them or invent new labels.  \n- for example: \"Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.\"",
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          "quadratic_profit": 101
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        {
          "item_i_id": "G",
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        },
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        {
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        {
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        },
        {
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        {
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        {
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        {
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          "quadratic_profit": 101
        },
        {
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          "quadratic_profit": 101
        },
        {
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          "item_j_id": "M",
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        {
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          "item_j_id": "N",
          "quadratic_profit": 101
        },
        {
          "item_i_id": "M",
          "item_j_id": "Q",
          "quadratic_profit": 101
        },
        {
          "item_i_id": "M",
          "item_j_id": "T",
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        },
        {
          "item_i_id": "N",
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          "quadratic_profit": 102
        },
        {
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        },
        {
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        },
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        },
        {
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        },
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        },
        {
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        {
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        },
        {
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        },
        {
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          "quadratic_profit": 102
        },
        {
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          "quadratic_profit": 101
        },
        {
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          "quadratic_profit": 102
        }
      ]
    },
    "solution_variant": [
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      "C",
      "F",
      "H",
      "I",
      "L",
      "M",
      "N",
      "Q",
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    "context_index": 39,
    "input_format": "json",
    "input_index_base": "names"
  },
  {
    "task_name": "QKP",
    "problem_type": "QKP",
    "instruction": "Recently the branch got a pile of donations and only so much shelf room to display them. Each book takes up some space and has its own circulation value, and some combinations of titles actually increase traffic when they’re displayed together. The plan is to pick which books to show so the shelf doesn’t overflow and the total payoff — the sum of the individual book benefits plus any extra pairwise bonuses when those two are both chosen — is maximized, with no duplicates allowed. The specific book sizes, benefits, and pairwise boosts follow below.\n\n{\n  \"num_books\": 16,\n  \"shelf_capacity\": 345,\n  \"book_ids\": [\n    1,\n    2,\n    3,\n    4,\n    5,\n    6,\n    7,\n    8,\n    9,\n    10,\n    11,\n    12,\n    13,\n    14,\n    15,\n    16\n  ],\n  \"linear\": [\n    {\n      \"book_id\": 1,\n      \"book_benefit\": 792\n    },\n    {\n      \"book_id\": 2,\n      \"book_benefit\": 238\n    },\n    {\n      \"book_id\": 3,\n      \"book_benefit\": 233\n    },\n    {\n      \"book_id\": 4,\n      \"book_benefit\": 259\n    },\n    {\n      \"book_id\": 5,\n      \"book_benefit\": 846\n    },\n    {\n      \"book_id\": 6,\n      \"book_benefit\": 136\n    },\n    {\n      \"book_id\": 7,\n      \"book_benefit\": 655\n    },\n    {\n      \"book_id\": 8,\n      \"book_benefit\": 341\n    },\n    {\n      \"book_id\": 9,\n      \"book_benefit\": 297\n    },\n    {\n      \"book_id\": 10,\n      \"book_benefit\": 418\n    },\n    {\n      \"book_id\": 11,\n      \"book_benefit\": 223\n    },\n    {\n      \"book_id\": 12,\n      \"book_benefit\": 809\n    },\n    {\n      \"book_id\": 13,\n      \"book_benefit\": 552\n    },\n    {\n      \"book_id\": 14,\n      \"book_benefit\": 823\n    },\n    {\n      \"book_id\": 15,\n      \"book_benefit\": 120\n    },\n    {\n      \"book_id\": 16,\n      \"book_benefit\": 420\n    }\n  ],\n  \"weights\": [\n    {\n      \"book_id\": 1,\n      \"book_size\": 715\n    },\n    {\n      \"book_id\": 2,\n      \"book_size\": 653\n    },\n    {\n      \"book_id\": 3,\n      \"book_size\": 160\n    },\n    {\n      \"book_id\": 4,\n      \"book_size\": 745\n    },\n    {\n      \"book_id\": 5,\n      \"book_size\": 518\n    },\n    {\n      \"book_id\": 6,\n      \"book_size\": 585\n    },\n    {\n      \"book_id\": 7,\n      \"book_size\": 733\n    },\n    {\n      \"book_id\": 8,\n      \"book_size\": 254\n    },\n    {\n      \"book_id\": 9,\n      \"book_size\": 513\n    },\n    {\n      \"book_id\": 10,\n      \"book_size\": 550\n    },\n    {\n      \"book_id\": 11,\n      \"book_size\": 476\n    },\n    {\n      \"book_id\": 12,\n      \"book_size\": 343\n    },\n    {\n      \"book_id\": 13,\n      \"book_size\": 864\n    },\n    {\n      \"book_id\": 14,\n      \"book_size\": 348\n    },\n    {\n      \"book_id\": 15,\n      \"book_size\": 327\n    },\n    {\n      \"book_id\": 16,\n      \"book_size\": 174\n    }\n  ],\n  \"quadratic\": [\n    {\n      \"book_i_id\": 1,\n      \"book_j_id\": 2,\n      \"pair_bonus\": 123\n    },\n    {\n      \"book_i_id\": 1,\n      \"book_j_id\": 11,\n      \"pair_bonus\": 110\n    },\n    {\n      \"book_i_id\": 1,\n      \"book_j_id\": 12,\n      \"pair_bonus\": 627\n    },\n    {\n      \"book_i_id\": 1,\n      \"book_j_id\": 13,\n      \"pair_bonus\": 567\n    },\n    {\n      \"book_i_id\": 1,\n      \"book_j_id\": 14,\n      \"pair_bonus\": 794\n    },\n    {\n      \"book_i_id\": 1,\n      \"book_j_id\": 16,\n      \"pair_bonus\": 656\n    },\n    {\n      \"book_i_id\": 2,\n      \"book_j_id\": 4,\n      \"pair_bonus\": 589\n    },\n    {\n      \"book_i_id\": 2,\n      \"book_j_id\": 7,\n      \"pair_bonus\": 448\n    },\n    {\n      \"book_i_id\": 2,\n      \"book_j_id\": 8,\n      \"pair_bonus\": 494\n    },\n    {\n      \"book_i_id\": 2,\n      \"book_j_id\": 10,\n      \"pair_bonus\": 470\n    },\n    {\n      \"book_i_id\": 2,\n      \"book_j_id\": 11,\n      \"pair_bonus\": 428\n    },\n    {\n      \"book_i_id\": 2,\n      \"book_j_id\": 12,\n      \"pair_bonus\": 877\n    },\n    {\n      \"book_i_id\": 2,\n      \"book_j_id\": 14,\n      \"pair_bonus\": 815\n    },\n    {\n      \"book_i_id\": 2,\n      \"book_j_id\": 16,\n      \"pair_bonus\": 115\n    },\n    {\n      \"book_i_id\": 3,\n      \"book_j_id\": 8,\n      \"pair_bonus\": 652\n    },\n    {\n      \"book_i_id\": 3,\n      \"book_j_id\": 9,\n      \"pair_bonus\": 712\n    },\n    {\n      \"book_i_id\": 3,\n      \"book_j_id\": 12,\n      \"pair_bonus\": 114\n    },\n    {\n      \"book_i_id\": 3,\n      \"book_j_id\": 14,\n      \"pair_bonus\": 442\n    },\n    {\n      \"book_i_id\": 3,\n      \"book_j_id\": 15,\n      \"pair_bonus\": 367\n    },\n    {\n      \"book_i_id\": 4,\n      \"book_j_id\": 8,\n      \"pair_bonus\": 604\n    },\n    {\n      \"book_i_id\": 4,\n      \"book_j_id\": 9,\n      \"pair_bonus\": 606\n    },\n    {\n      \"book_i_id\": 4,\n      \"book_j_id\": 11,\n      \"pair_bonus\": 417\n    },\n    {\n      \"book_i_id\": 4,\n      \"book_j_id\": 12,\n      \"pair_bonus\": 861\n    },\n    {\n      \"book_i_id\": 4,\n      \"book_j_id\": 15,\n      \"pair_bonus\": 372\n    },\n    {\n      \"book_i_id\": 5,\n      \"book_j_id\": 7,\n      \"pair_bonus\": 650\n    },\n    {\n      \"book_i_id\": 5,\n      \"book_j_id\": 8,\n      \"pair_bonus\": 773\n    },\n    {\n      \"book_i_id\": 5,\n      \"book_j_id\": 12,\n      \"pair_bonus\": 362\n    },\n    {\n      \"book_i_id\": 5,\n      \"book_j_id\": 13,\n      \"pair_bonus\": 812\n    },\n    {\n      \"book_i_id\": 5,\n      \"book_j_id\": 14,\n      \"pair_bonus\": 245\n    },\n    {\n      \"book_i_id\": 5,\n      \"book_j_id\": 15,\n      \"pair_bonus\": 674\n    },\n    {\n      \"book_i_id\": 6,\n      \"book_j_id\": 7,\n      \"pair_bonus\": 765\n    },\n    {\n      \"book_i_id\": 6,\n      \"book_j_id\": 11,\n      \"pair_bonus\": 865\n    },\n    {\n      \"book_i_id\": 6,\n      \"book_j_id\": 13,\n      \"pair_bonus\": 610\n    },\n    {\n      \"book_i_id\": 6,\n      \"book_j_id\": 16,\n      \"pair_bonus\": 562\n    },\n    {\n      \"book_i_id\": 7,\n      \"book_j_id\": 9,\n      \"pair_bonus\": 855\n    },\n    {\n      \"book_i_id\": 7,\n      \"book_j_id\": 10,\n      \"pair_bonus\": 168\n    },\n    {\n      \"book_i_id\": 7,\n      \"book_j_id\": 12,\n      \"pair_bonus\": 254\n    },\n    {\n      \"book_i_id\": 7,\n      \"book_j_id\": 15,\n      \"pair_bonus\": 575\n    },\n    {\n      \"book_i_id\": 8,\n      \"book_j_id\": 9,\n      \"pair_bonus\": 135\n    },\n    {\n      \"book_i_id\": 8,\n      \"book_j_id\": 10,\n      \"pair_bonus\": 774\n    },\n    {\n      \"book_i_id\": 8,\n      \"book_j_id\": 11,\n      \"pair_bonus\": 762\n    },\n    {\n      \"book_i_id\": 8,\n      \"book_j_id\": 14,\n      \"pair_bonus\": 345\n    },\n    {\n      \"book_i_id\": 8,\n      \"book_j_id\": 15,\n      \"pair_bonus\": 464\n    },\n    {\n      \"book_i_id\": 8,\n      \"book_j_id\": 16,\n      \"pair_bonus\": 494\n    },\n    {\n      \"book_i_id\": 9,\n      \"book_j_id\": 14,\n      \"pair_bonus\": 384\n    },\n    {\n      \"book_i_id\": 9,\n      \"book_j_id\": 15,\n      \"pair_bonus\": 616\n    },\n    {\n      \"book_i_id\": 10,\n      \"book_j_id\": 11,\n      \"pair_bonus\": 333\n    },\n    {\n      \"book_i_id\": 10,\n      \"book_j_id\": 12,\n      \"pair_bonus\": 851\n    },\n    {\n      \"book_i_id\": 10,\n      \"book_j_id\": 13,\n      \"pair_bonus\": 141\n    },\n    {\n      \"book_i_id\": 10,\n      \"book_j_id\": 15,\n      \"pair_bonus\": 277\n    },\n    {\n      \"book_i_id\": 10,\n      \"book_j_id\": 16,\n      \"pair_bonus\": 634\n    },\n    {\n      \"book_i_id\": 11,\n      \"book_j_id\": 12,\n      \"pair_bonus\": 528\n    },\n    {\n      \"book_i_id\": 12,\n      \"book_j_id\": 13,\n      \"pair_bonus\": 661\n    },\n    {\n      \"book_i_id\": 12,\n      \"book_j_id\": 14,\n      \"pair_bonus\": 699\n    },\n    {\n      \"book_i_id\": 13,\n      \"book_j_id\": 16,\n      \"pair_bonus\": 297\n    },\n    {\n      \"book_i_id\": 14,\n      \"book_j_id\": 16,\n      \"pair_bonus\": 213\n    }\n  ]\n}\n\nWhen you send your pick, just stick to this simple JSON layout so I know which books you chose — nothing fancy, just the list of book ids.\n\n{\n  \"solution\": [<book_id>, <book_id>, ...]\n}\n\nPretty straightforward: \"solution\" is the list of the books you want to put on the shelf, each entry being the book's identifier from the instance. Think of it like filling in a short form that names the books you’re displaying.\n\nThis JSON is only a sketch of the shape I expect you to return — not the final answer itself.\n\nPlease make sure to use the identifiers exactly as they appear in the instance input — no renaming, and don’t introduce any new labels.\n\nValid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.",
    "instance": {
      "linear_coeffs": [
        792,
        238,
        233,
        259,
        846,
        136,
        655,
        341,
        297,
        418,
        223,
        809,
        552,
        823,
        120,
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      ],
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        [
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          494,
          0,
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        ],
        [
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        [
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        ],
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        ],
        [
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        ],
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      ],
      "weights": [
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        343,
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      ],
      "capacity": 345,
      "solution": [
        11
      ],
      "obj": 809.0,
      "problem_type": "QKP"
    },
    "solution": [
      11
    ],
    "obj": 809.0,
    "instance_variant": {
      "problem_type": "QKP",
      "num_items": 16,
      "capacity": 345,
      "items": [
        1,
        2,
        3,
        4,
        5,
        6,
        7,
        8,
        9,
        10,
        11,
        12,
        13,
        14,
        15,
        16
      ],
      "linear_pairs": [
        {
          "item_id": 1,
          "linear_profit": 792
        },
        {
          "item_id": 2,
          "linear_profit": 238
        },
        {
          "item_id": 3,
          "linear_profit": 233
        },
        {
          "item_id": 4,
          "linear_profit": 259
        },
        {
          "item_id": 5,
          "linear_profit": 846
        },
        {
          "item_id": 6,
          "linear_profit": 136
        },
        {
          "item_id": 7,
          "linear_profit": 655
        },
        {
          "item_id": 8,
          "linear_profit": 341
        },
        {
          "item_id": 9,
          "linear_profit": 297
        },
        {
          "item_id": 10,
          "linear_profit": 418
        },
        {
          "item_id": 11,
          "linear_profit": 223
        },
        {
          "item_id": 12,
          "linear_profit": 809
        },
        {
          "item_id": 13,
          "linear_profit": 552
        },
        {
          "item_id": 14,
          "linear_profit": 823
        },
        {
          "item_id": 15,
          "linear_profit": 120
        },
        {
          "item_id": 16,
          "linear_profit": 420
        }
      ],
      "weight_pairs": [
        {
          "item_id": 1,
          "weight": 715
        },
        {
          "item_id": 2,
          "weight": 653
        },
        {
          "item_id": 3,
          "weight": 160
        },
        {
          "item_id": 4,
          "weight": 745
        },
        {
          "item_id": 5,
          "weight": 518
        },
        {
          "item_id": 6,
          "weight": 585
        },
        {
          "item_id": 7,
          "weight": 733
        },
        {
          "item_id": 8,
          "weight": 254
        },
        {
          "item_id": 9,
          "weight": 513
        },
        {
          "item_id": 10,
          "weight": 550
        },
        {
          "item_id": 11,
          "weight": 476
        },
        {
          "item_id": 12,
          "weight": 343
        },
        {
          "item_id": 13,
          "weight": 864
        },
        {
          "item_id": 14,
          "weight": 348
        },
        {
          "item_id": 15,
          "weight": 327
        },
        {
          "item_id": 16,
          "weight": 174
        }
      ],
      "quadratic_pairs": [
        {
          "item_i_id": 1,
          "item_j_id": 2,
          "quadratic_profit": 123
        },
        {
          "item_i_id": 1,
          "item_j_id": 11,
          "quadratic_profit": 110
        },
        {
          "item_i_id": 1,
          "item_j_id": 12,
          "quadratic_profit": 627
        },
        {
          "item_i_id": 1,
          "item_j_id": 13,
          "quadratic_profit": 567
        },
        {
          "item_i_id": 1,
          "item_j_id": 14,
          "quadratic_profit": 794
        },
        {
          "item_i_id": 1,
          "item_j_id": 16,
          "quadratic_profit": 656
        },
        {
          "item_i_id": 2,
          "item_j_id": 2,
          "quadratic_profit": 402
        },
        {
          "item_i_id": 2,
          "item_j_id": 4,
          "quadratic_profit": 589
        },
        {
          "item_i_id": 2,
          "item_j_id": 7,
          "quadratic_profit": 448
        },
        {
          "item_i_id": 2,
          "item_j_id": 8,
          "quadratic_profit": 494
        },
        {
          "item_i_id": 2,
          "item_j_id": 10,
          "quadratic_profit": 470
        },
        {
          "item_i_id": 2,
          "item_j_id": 11,
          "quadratic_profit": 428
        },
        {
          "item_i_id": 2,
          "item_j_id": 12,
          "quadratic_profit": 877
        },
        {
          "item_i_id": 2,
          "item_j_id": 14,
          "quadratic_profit": 815
        },
        {
          "item_i_id": 2,
          "item_j_id": 16,
          "quadratic_profit": 115
        },
        {
          "item_i_id": 3,
          "item_j_id": 8,
          "quadratic_profit": 652
        },
        {
          "item_i_id": 3,
          "item_j_id": 9,
          "quadratic_profit": 712
        },
        {
          "item_i_id": 3,
          "item_j_id": 12,
          "quadratic_profit": 114
        },
        {
          "item_i_id": 3,
          "item_j_id": 14,
          "quadratic_profit": 442
        },
        {
          "item_i_id": 3,
          "item_j_id": 15,
          "quadratic_profit": 367
        },
        {
          "item_i_id": 4,
          "item_j_id": 8,
          "quadratic_profit": 604
        },
        {
          "item_i_id": 4,
          "item_j_id": 9,
          "quadratic_profit": 606
        },
        {
          "item_i_id": 4,
          "item_j_id": 11,
          "quadratic_profit": 417
        },
        {
          "item_i_id": 4,
          "item_j_id": 12,
          "quadratic_profit": 861
        },
        {
          "item_i_id": 4,
          "item_j_id": 15,
          "quadratic_profit": 372
        },
        {
          "item_i_id": 5,
          "item_j_id": 5,
          "quadratic_profit": 233
        },
        {
          "item_i_id": 5,
          "item_j_id": 7,
          "quadratic_profit": 650
        },
        {
          "item_i_id": 5,
          "item_j_id": 8,
          "quadratic_profit": 773
        },
        {
          "item_i_id": 5,
          "item_j_id": 12,
          "quadratic_profit": 362
        },
        {
          "item_i_id": 5,
          "item_j_id": 13,
          "quadratic_profit": 812
        },
        {
          "item_i_id": 5,
          "item_j_id": 14,
          "quadratic_profit": 245
        },
        {
          "item_i_id": 5,
          "item_j_id": 15,
          "quadratic_profit": 674
        },
        {
          "item_i_id": 6,
          "item_j_id": 6,
          "quadratic_profit": 657
        },
        {
          "item_i_id": 6,
          "item_j_id": 7,
          "quadratic_profit": 765
        },
        {
          "item_i_id": 6,
          "item_j_id": 11,
          "quadratic_profit": 865
        },
        {
          "item_i_id": 6,
          "item_j_id": 13,
          "quadratic_profit": 610
        },
        {
          "item_i_id": 6,
          "item_j_id": 16,
          "quadratic_profit": 562
        },
        {
          "item_i_id": 7,
          "item_j_id": 9,
          "quadratic_profit": 855
        },
        {
          "item_i_id": 7,
          "item_j_id": 10,
          "quadratic_profit": 168
        },
        {
          "item_i_id": 7,
          "item_j_id": 12,
          "quadratic_profit": 254
        },
        {
          "item_i_id": 7,
          "item_j_id": 15,
          "quadratic_profit": 575
        },
        {
          "item_i_id": 8,
          "item_j_id": 9,
          "quadratic_profit": 135
        },
        {
          "item_i_id": 8,
          "item_j_id": 10,
          "quadratic_profit": 774
        },
        {
          "item_i_id": 8,
          "item_j_id": 11,
          "quadratic_profit": 762
        },
        {
          "item_i_id": 8,
          "item_j_id": 14,
          "quadratic_profit": 345
        },
        {
          "item_i_id": 8,
          "item_j_id": 15,
          "quadratic_profit": 464
        },
        {
          "item_i_id": 8,
          "item_j_id": 16,
          "quadratic_profit": 494
        },
        {
          "item_i_id": 9,
          "item_j_id": 9,
          "quadratic_profit": 881
        },
        {
          "item_i_id": 9,
          "item_j_id": 14,
          "quadratic_profit": 384
        },
        {
          "item_i_id": 9,
          "item_j_id": 15,
          "quadratic_profit": 616
        },
        {
          "item_i_id": 10,
          "item_j_id": 11,
          "quadratic_profit": 333
        },
        {
          "item_i_id": 10,
          "item_j_id": 12,
          "quadratic_profit": 851
        },
        {
          "item_i_id": 10,
          "item_j_id": 13,
          "quadratic_profit": 141
        },
        {
          "item_i_id": 10,
          "item_j_id": 15,
          "quadratic_profit": 277
        },
        {
          "item_i_id": 10,
          "item_j_id": 16,
          "quadratic_profit": 634
        },
        {
          "item_i_id": 11,
          "item_j_id": 11,
          "quadratic_profit": 867
        },
        {
          "item_i_id": 11,
          "item_j_id": 12,
          "quadratic_profit": 528
        },
        {
          "item_i_id": 12,
          "item_j_id": 12,
          "quadratic_profit": 327
        },
        {
          "item_i_id": 12,
          "item_j_id": 13,
          "quadratic_profit": 661
        },
        {
          "item_i_id": 12,
          "item_j_id": 14,
          "quadratic_profit": 699
        },
        {
          "item_i_id": 13,
          "item_j_id": 13,
          "quadratic_profit": 233
        },
        {
          "item_i_id": 13,
          "item_j_id": 16,
          "quadratic_profit": 297
        },
        {
          "item_i_id": 14,
          "item_j_id": 14,
          "quadratic_profit": 762
        },
        {
          "item_i_id": 14,
          "item_j_id": 16,
          "quadratic_profit": 213
        }
      ]
    },
    "solution_variant": [
      12
    ],
    "context_index": 40,
    "input_format": "json",
    "input_index_base": 1
  },
  {
    "task_name": "QKP",
    "problem_type": "QKP",
    "instruction": "I’m prepping for a pop-up plant sale and trying to pack a single flatbed without going over its maximum load, so there’s a decision to make about which pots to bring. Every pot contributes a base margin, and a few specific pairings add extra margin if both make the trip. The aim is to pick the set of plants whose combined weight stays under the limit and that gives the largest possible take — by summing up each chosen pot’s margin plus any extra bonuses for those special pairs — and each plant is either taken once or left. The concrete details (plant list, weights, margins, pair bonuses) are given below.\n\nBelow are the 23 plants (A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V, W) and the flatbed max load 252.\nI can include A for a base margin of 234.\nI can include B for a base margin of 160.\nI can include C for a base margin of 185.\nI can include D for a base margin of 157.\nI can include E for a base margin of 171.\nI can include F for a base margin of 129.\nI can include G for a base margin of 174.\nI can include H for a base margin of 271.\nI can include I for a base margin of 127.\nI can include J for a base margin of 128.\nI can include K for a base margin of 217.\nI can include L for a base margin of 268.\nI can include M for a base margin of 95.\nI can include N for a base margin of 260.\nI can include O for a base margin of 189.\nI can include P for a base margin of 148.\nI can include Q for a base margin of 108.\nI can include R for a base margin of 194.\nI can include S for a base margin of 98.\nI can include T for a base margin of 265.\nI can include U for a base margin of 247.\nI can include V for a base margin of 203.\nI can include W for a base margin of 185.\nI can include A which weighs 235.\nI can include B which weighs 146.\nI can include C which weighs 199.\nI can include D which weighs 166.\nI can include E which weighs 185.\nI can include F which weighs 131.\nI can include G which weighs 177.\nI can include H which weighs 274.\nI can include I which weighs 129.\nI can include J which weighs 142.\nI can include K which weighs 211.\nI can include L which weighs 271.\nI can include M which weighs 102.\nI can include N which weighs 253.\nI can include O which weighs 201.\nI can include P which weighs 160.\nI can include Q which weighs 106.\nI can include R which weighs 184.\nI can include S which weighs 107.\nI can include T which weighs 270.\nI can include U which weighs 234.\nI can include V which weighs 190.\nI can include W which weighs 198.\nIf I load A together with B, I earn an extra 241.\nIf I load A together with C, I earn an extra 199.\nIf I load A together with F, I earn an extra 253.\nIf I load A together with G, I earn an extra 242.\nIf I load A together with I, I earn an extra 232.\nIf I load A together with J, I earn an extra 223.\nIf I load A together with L, I earn an extra 231.\nIf I load A together with M, I earn an extra 162.\nIf I load A together with N, I earn an extra 203.\nIf I load A together with O, I earn an extra 172.\nIf I load A together with P, I earn an extra 251.\nIf I load A together with Q, I earn an extra 200.\nIf I load A together with R, I earn an extra 247.\nIf I load A together with S, I earn an extra 248.\nIf I load A together with T, I earn an extra 198.\nIf I load A together with U, I earn an extra 264.\nIf I load A together with W, I earn an extra 219.\nIf I load B together with D, I earn an extra 199.\nIf I load B together with E, I earn an extra 160.\nIf I load B together with F, I earn an extra 152.\nIf I load B together with G, I earn an extra 124.\nIf I load B together with H, I earn an extra 126.\nIf I load B together with I, I earn an extra 200.\nIf I load B together with J, I earn an extra 154.\nIf I load B together with K, I earn an extra 156.\nIf I load B together with M, I earn an extra 158.\nIf I load B together with N, I earn an extra 162.\nIf I load B together with O, I earn an extra 166.\nIf I load B together with P, I earn an extra 175.\nIf I load B together with Q, I earn an extra 163.\nIf I load B together with R, I earn an extra 153.\nIf I load B together with S, I earn an extra 184.\nIf I load B together with T, I earn an extra 173.\nIf I load B together with U, I earn an extra 166.\nIf I load B together with W, I earn an extra 174.\nIf I load C together with D, I earn an extra 233.\nIf I load C together with G, I earn an extra 213.\nIf I load C together with I, I earn an extra 158.\nIf I load C together with J, I earn an extra 199.\nIf I load C together with K, I earn an extra 231.\nIf I load C together with L, I earn an extra 180.\nIf I load C together with M, I earn an extra 191.\nIf I load C together with N, I earn an extra 198.\nIf I load C together with O, I earn an extra 243.\nIf I load C together with P, I earn an extra 201.\nIf I load C together with Q, I earn an extra 237.\nIf I load C together with S, I earn an extra 164.\nIf I load C together with V, I earn an extra 124.\nIf I load C together with W, I earn an extra 164.\nIf I load D together with E, I earn an extra 222.\nIf I load D together with G, I earn an extra 148.\nIf I load D together with H, I earn an extra 177.\nIf I load D together with I, I earn an extra 185.\nIf I load D together with J, I earn an extra 207.\nIf I load D together with K, I earn an extra 186.\nIf I load D together with M, I earn an extra 184.\nIf I load D together with N, I earn an extra 180.\nIf I load D together with P, I earn an extra 147.\nIf I load D together with Q, I earn an extra 185.\nIf I load D together with R, I earn an extra 231.\nIf I load D together with S, I earn an extra 205.\nIf I load D together with T, I earn an extra 265.\nIf I load D together with V, I earn an extra 204.\nIf I load D together with W, I earn an extra 186.\nIf I load E together with G, I earn an extra 177.\nIf I load E together with H, I earn an extra 194.\nIf I load E together with I, I earn an extra 153.\nIf I load E together with J, I earn an extra 209.\nIf I load E together with K, I earn an extra 191.\nIf I load E together with L, I earn an extra 228.\nIf I load E together with M, I earn an extra 190.\nIf I load E together with O, I earn an extra 159.\nIf I load E together with P, I earn an extra 198.\nIf I load E together with Q, I earn an extra 218.\nIf I load E together with R, I earn an extra 202.\nIf I load E together with S, I earn an extra 252.\nIf I load E together with T, I earn an extra 172.\nIf I load E together with U, I earn an extra 159.\nIf I load E together with V, I earn an extra 246.\nIf I load E together with W, I earn an extra 239.\nIf I load F together with G, I earn an extra 170.\nIf I load F together with H, I earn an extra 143.\nIf I load F together with I, I earn an extra 185.\nIf I load F together with J, I earn an extra 145.\nIf I load F together with L, I earn an extra 131.\nIf I load F together with M, I earn an extra 165.\nIf I load F together with N, I earn an extra 183.\nIf I load F together with P, I earn an extra 257.\nIf I load F together with Q, I earn an extra 228.\nIf I load F together with S, I earn an extra 227.\nIf I load F together with T, I earn an extra 200.\nIf I load F together with V, I earn an extra 167.\nIf I load F together with W, I earn an extra 247.\nIf I load G together with H, I earn an extra 215.\nIf I load G together with I, I earn an extra 171.\nIf I load G together with J, I earn an extra 146.\nIf I load G together with L, I earn an extra 247.\nIf I load G together with M, I earn an extra 267.\nIf I load G together with N, I earn an extra 292.\nIf I load G together with O, I earn an extra 266.\nIf I load G together with Q, I earn an extra 174.\nIf I load G together with R, I earn an extra 187.\nIf I load G together with S, I earn an extra 228.\nIf I load G together with U, I earn an extra 174.\nIf I load H together with I, I earn an extra 210.\nIf I load H together with K, I earn an extra 239.\nIf I load H together with L, I earn an extra 166.\nIf I load H together with M, I earn an extra 185.\nIf I load H together with N, I earn an extra 169.\nIf I load H together with O, I earn an extra 174.\nIf I load H together with P, I earn an extra 237.\nIf I load H together with Q, I earn an extra 230.\nIf I load H together with R, I earn an extra 253.\nIf I load H together with S, I earn an extra 157.\nIf I load H together with U, I earn an extra 201.\nIf I load H together with V, I earn an extra 209.\nIf I load H together with W, I earn an extra 174.\nIf I load I together with J, I earn an extra 142.\nIf I load I together with K, I earn an extra 130.\nIf I load I together with L, I earn an extra 198.\nIf I load I together with M, I earn an extra 184.\nIf I load I together with O, I earn an extra 193.\nIf I load I together with Q, I earn an extra 161.\nIf I load I together with T, I earn an extra 150.\nIf I load I together with U, I earn an extra 127.\nIf I load I together with W, I earn an extra 203.\nIf I load J together with N, I earn an extra 172.\nIf I load J together with O, I earn an extra 139.\nIf I load J together with Q, I earn an extra 117.\nIf I load J together with R, I earn an extra 130.\nIf I load J together with S, I earn an extra 158.\nIf I load J together with T, I earn an extra 123.\nIf I load J together with U, I earn an extra 151.\nIf I load J together with V, I earn an extra 146.\nIf I load K together with L, I earn an extra 132.\nIf I load K together with M, I earn an extra 194.\nIf I load K together with N, I earn an extra 219.\nIf I load K together with P, I earn an extra 164.\nIf I load K together with Q, I earn an extra 173.\nIf I load K together with R, I earn an extra 160.\nIf I load K together with S, I earn an extra 199.\nIf I load K together with T, I earn an extra 185.\nIf I load K together with U, I earn an extra 177.\nIf I load K together with V, I earn an extra 205.\nIf I load L together with M, I earn an extra 129.\nIf I load L together with N, I earn an extra 128.\nIf I load L together with P, I earn an extra 145.\nIf I load L together with Q, I earn an extra 144.\nIf I load L together with T, I earn an extra 110.\nIf I load L together with U, I earn an extra 152.\nIf I load L together with V, I earn an extra 154.\nIf I load L together with W, I earn an extra 148.\nIf I load M together with N, I earn an extra 126.\nIf I load M together with O, I earn an extra 141.\nIf I load M together with P, I earn an extra 181.\nIf I load M together with T, I earn an extra 182.\nIf I load M together with U, I earn an extra 219.\nIf I load M together with W, I earn an extra 231.\nIf I load N together with O, I earn an extra 230.\nIf I load N together with P, I earn an extra 282.\nIf I load N together with Q, I earn an extra 207.\nIf I load N together with R, I earn an extra 191.\nIf I load N together with S, I earn an extra 152.\nIf I load N together with T, I earn an extra 179.\nIf I load N together with U, I earn an extra 152.\nIf I load N together with V, I earn an extra 229.\nIf I load N together with W, I earn an extra 244.\nIf I load O together with P, I earn an extra 146.\nIf I load O together with Q, I earn an extra 130.\nIf I load O together with R, I earn an extra 130.\nIf I load O together with S, I earn an extra 208.\nIf I load O together with T, I earn an extra 210.\nIf I load O together with U, I earn an extra 178.\nIf I load O together with V, I earn an extra 143.\nIf I load O together with W, I earn an extra 223.\nIf I load P together with Q, I earn an extra 159.\nIf I load P together with S, I earn an extra 164.\nIf I load P together with T, I earn an extra 172.\nIf I load P together with V, I earn an extra 193.\nIf I load P together with W, I earn an extra 141.\nIf I load Q together with R, I earn an extra 147.\nIf I load Q together with S, I earn an extra 136.\nIf I load Q together with T, I earn an extra 142.\nIf I load Q together with V, I earn an extra 122.\nIf I load R together with S, I earn an extra 148.\nIf I load R together with T, I earn an extra 148.\nIf I load R together with V, I earn an extra 246.\nIf I load R together with W, I earn an extra 169.\nIf I load S together with T, I earn an extra 253.\nIf I load S together with U, I earn an extra 235.\nIf I load S together with W, I earn an extra 217.\nIf I load T together with U, I earn an extra 223.\nIf I load T together with V, I earn an extra 185.\nIf I load T together with W, I earn an extra 185.\nIf I load U together with V, I earn an extra 208.\nIf I load U together with W, I earn an extra 229.\nIf I load V together with W, I earn an extra 141.\nI'll use these details to pick plants that stay under 252 and maximize my take.\n\nAlso, when you give your pick, please put it in a tiny JSON snippet like this so it's easy to read and check:\n\n{\n  \"solution\": [\"<plant_id>\", \"<plant_id>\", ...]\n}\n\nHere \"solution\" is just a list of the plant identifiers you're planning to load onto the flatbed. Think of it as the simple answer form: the names (IDs) of the pots you’ll bring, nothing else. This is just a sketch of the shape I expect, not the actual packing decision.\n\nPlease use the exact identifiers from the instance input — no renaming and no new labels. \n- for example: \"Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.\"",
    "instance": {
      "linear_coeffs": [
        234,
        160,
        185,
        157,
        171,
        129,
        174,
        271,
        127,
        128,
        217,
        268,
        95,
        260,
        189,
        148,
        108,
        194,
        98,
        265,
        247,
        203,
        185
      ],
      "quadratic_coeffs": [
        [
          238,
          241,
          199,
          0,
          0,
          253,
          242,
          0,
          232,
          223,
          0,
          231,
          162,
          203,
          172,
          251,
          200,
          247,
          248,
          198,
          264,
          0,
          219
        ],
        [
          0,
          148,
          0,
          199,
          160,
          152,
          124,
          126,
          200,
          154,
          156,
          0,
          158,
          162,
          166,
          175,
          163,
          153,
          184,
          173,
          166,
          0,
          174
        ],
        [
          0,
          0,
          141,
          233,
          0,
          0,
          213,
          0,
          158,
          199,
          231,
          180,
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        {
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        {
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        {
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        },
        {
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        },
        {
          "item_i_id": "K",
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        },
        {
          "item_i_id": "K",
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        {
          "item_i_id": "L",
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        {
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        {
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        {
          "item_i_id": "N",
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        {
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        {
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        {
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        {
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        {
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        {
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        },
        {
          "item_i_id": "S",
          "item_j_id": "W",
          "quadratic_profit": 217
        },
        {
          "item_i_id": "T",
          "item_j_id": "U",
          "quadratic_profit": 223
        },
        {
          "item_i_id": "T",
          "item_j_id": "V",
          "quadratic_profit": 185
        },
        {
          "item_i_id": "T",
          "item_j_id": "W",
          "quadratic_profit": 185
        },
        {
          "item_i_id": "U",
          "item_j_id": "U",
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        },
        {
          "item_i_id": "U",
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        },
        {
          "item_i_id": "U",
          "item_j_id": "W",
          "quadratic_profit": 229
        },
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          "item_j_id": "V",
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        },
        {
          "item_i_id": "V",
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        },
        {
          "item_i_id": "W",
          "item_j_id": "W",
          "quadratic_profit": 195
        }
      ]
    },
    "solution_variant": [
      "F",
      "Q"
    ],
    "context_index": 41,
    "input_format": "nl",
    "input_index_base": "names"
  },
  {
    "task_name": "QKP",
    "problem_type": "QKP",
    "instruction": "I’m hauling a crate of vinyl to a gig and need to pick what goes in it. Each record has its own crowd-pleasing score, and certain pairs of records boost the vibe even more if both are in the set — the total payoff is the sum of each record’s score plus any extra bonuses from pairs that end up together. The catch is the crate can only hold so much weight and there’s only one copy of each record, so the aim is to get the highest total score while keeping the total weight under the transport limit. The exact records, weights, and pair bonuses are listed below.\n\nI'm choosing from 23 records: A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V, W, and my crate holds up to 2733 weight — details below.\n\n| record_id | crowd_pleasing_score |\n|---|---|\n| A | 118 |\n| B | 217 |\n| C | 123 |\n| D | 165 |\n| E | 180 |\n| F | 231 |\n| G | 105 |\n| H | 105 |\n| I | 141 |\n| J | 226 |\n| K | 203 |\n| L | 274 |\n| M | 208 |\n| N | 162 |\n| O | 199 |\n| P | 267 |\n| Q | 208 |\n| R | 235 |\n| S | 239 |\n| T | 266 |\n| U | 196 |\n| V | 255 |\n| W | 301 |\n\n| record_id | record_weight |\n|---|---|\n| A | 122 |\n| B | 221 |\n| C | 133 |\n| D | 154 |\n| E | 171 |\n| F | 233 |\n| G | 101 |\n| H | 102 |\n| I | 132 |\n| J | 214 |\n| K | 205 |\n| L | 274 |\n| M | 223 |\n| N | 161 |\n| O | 201 |\n| P | 275 |\n| Q | 214 |\n| R | 246 |\n| S | 235 |\n| T | 279 |\n| U | 194 |\n| V | 253 |\n| W | 294 |\n\n| record_i_id | record_j_id | pair_synergy_bonus |\n|---|---|---|\n| A | H | 122 |\n| A | J | 122 |\n| A | L | 122 |\n| A | O | 103 |\n| A | P | 122 |\n| A | Q | 122 |\n| A | R | 122 |\n| A | U | 122 |\n| A | V | 114 |\n| A | W | 147 |\n| B | D | 148 |\n| B | E | 165 |\n| B | I | 174 |\n| B | J | 122 |\n| B | K | 221 |\n| B | N | 103 |\n| B | V | 163 |\n| C | Q | 133 |\n| C | W | 191 |\n| D | G | 154 |\n| D | H | 154 |\n| D | I | 133 |\n| D | K | 154 |\n| D | M | 154 |\n| D | U | 153 |\n| D | V | 153 |\n| E | G | 114 |\n| E | I | 171 |\n| E | M | 123 |\n| E | N | 171 |\n| E | T | 185 |\n| F | I | 186 |\n| F | J | 213 |\n| F | P | 233 |\n| F | R | 224 |\n| F | U | 165 |\n| F | W | 165 |\n| G | L | 101 |\n| G | P | 101 |\n| G | R | 101 |\n| H | I | 102 |\n| H | K | 102 |\n| H | O | 102 |\n| H | T | 245 |\n| H | U | 245 |\n| I | K | 132 |\n| I | O | 132 |\n| I | T | 199 |\n| J | N | 164 |\n| J | O | 107 |\n| J | U | 111 |\n| K | O | 202 |\n| L | T | 114 |\n| M | O | 156 |\n| M | P | 149 |\n| N | Q | 144 |\n| N | T | 132 |\n| O | S | 151 |\n| O | U | 186 |\n| O | W | 164 |\n| P | V | 137 |\n| R | S | 114 |\n| S | V | 131 |\n| T | W | 119 |\n| U | V | 184 |\n\nI'll aim to pack the best-scoring mix without exceeding 2733.\n\nAlso, when you tell me which records make the cut, a tiny JSON snippet like this is the easiest way to send it back — nice and simple:\n\n{\n  \"solution\": [<record_id>, <record_id>, ...]\n}\n\nHere, \"solution\" is just a list of the record IDs you want in the crate — nothing fancy, just the exact identifiers of the records you picked. Think of it as the checklist: each placeholder like <record_id> stands for one record you include.\n\nThis is just the shape I need, not your final score or any calculations — just the chosen IDs. Please use the item identifiers exactly as they appear in the instance input — do not rename them or invent new labels.\n\n- for example: \"Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.\"",
    "instance": {
      "linear_coeffs": [
        118,
        217,
        123,
        165,
        180,
        231,
        105,
        105,
        141,
        226,
        203,
        274,
        208,
        162,
        199,
        267,
        208,
        235,
        239,
        266,
        196,
        255,
        301
      ],
      "quadratic_coeffs": [
        [
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          0,
          0,
          0,
          0,
          0,
          0,
          122,
          0,
          122,
          0,
          122,
          0,
          0,
          103,
          122,
          122,
          122,
          0,
          0,
          122,
          114,
          147
        ],
        [
          0,
          191,
          0,
          148,
          165,
          0,
          0,
          0,
          174,
          122,
          221,
          0,
          0,
          103,
          0,
          0,
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        ],
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          0,
          0,
          0,
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          0,
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        ],
        [
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          0,
          0,
          0,
          0,
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          154,
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          154,
          0,
          154,
          0,
          0,
          0,
          0,
          0,
          0,
          0,
          153,
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          0
        ],
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          0,
          0,
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          0,
          0,
          114,
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          0,
          0,
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          123,
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          0,
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        ],
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          0,
          165
        ],
        [
          0,
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          0,
          0,
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          0,
          0,
          0,
          0,
          0,
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          0
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        [
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          102,
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    "instruction": "We picture an aid team packing a single cargo box: there are a bunch of different aid kits, each with its own weight and a base amount of good it will do once delivered. On top of that, certain combinations of two kits create extra effectiveness when both travel together. The task is simple in idea — try to get as much total relief as possible by adding up each selected kit’s base impact and any extra pairwise bonuses — but stay within the box’s weight limit. Only whole kits are available and you can’t pick the same kit twice. The exact weights, impacts, and combo bonuses appear below.\n\n# num_kits=24\n# container_capacity=356\n# kit_ids=0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23\n\nkit_id,base_impact\n0,99\n1,105\n2,103\n3,103\n4,105\n5,103\n6,104\n7,106\n8,102\n9,101\n10,101\n11,97\n12,99\n13,96\n14,108\n15,101\n16,99\n17,103\n18,96\n19,102\n20,110\n21,104\n22,102\n23,101\n\nkit_id,kit_weight\n0,100\n1,100\n2,103\n3,103\n4,103\n5,102\n6,105\n7,101\n8,100\n9,101\n10,100\n11,100\n12,101\n13,100\n14,103\n15,100\n16,104\n17,105\n18,100\n19,103\n20,105\n21,102\n22,104\n23,105\n\nkit_i_id,kit_j_id,pairwise_boost\n0,4,100\n0,5,100\n0,8,100\n0,9,100\n0,10,100\n0,11,100\n0,16,100\n0,18,100\n0,21,100\n1,5,100\n1,11,100\n1,13,100\n1,17,100\n1,20,100\n1,22,101\n2,4,103\n2,5,103\n2,6,102\n2,11,103\n2,13,103\n2,14,102\n2,15,103\n2,16,100\n2,17,103\n2,19,103\n2,21,102\n3,4,101\n3,5,103\n3,6,103\n3,7,103\n3,8,103\n3,9,103\n3,10,102\n3,13,103\n3,14,103\n3,15,103\n3,19,101\n3,20,104\n3,21,100\n3,22,101\n4,5,103\n4,6,103\n4,8,101\n4,10,101\n4,11,103\n4,15,101\n4,16,100\n4,18,102\n4,19,102\n4,20,102\n4,22,102\n4,23,101\n5,7,101\n5,11,102\n5,12,102\n5,13,102\n5,14,102\n5,15,102\n5,17,101\n5,19,100\n5,21,100\n5,22,100\n6,7,100\n6,9,103\n6,14,100\n6,15,103\n6,17,100\n6,20,101\n7,13,101\n7,14,101\n7,16,100\n7,19,100\n7,20,100\n7,23,100\n8,10,100\n8,12,100\n8,13,100\n8,14,100\n8,18,101\n8,19,104\n8,21,102\n8,23,104\n9,10,101\n9,11,100\n9,12,101\n9,14,101\n9,17,103\n9,19,103\n9,21,101\n9,23,103\n10,12,100\n10,13,103\n10,14,100\n10,15,105\n10,17,105\n10,18,103\n10,20,104\n10,21,100\n10,22,101\n10,23,105\n11,12,100\n11,13,100\n11,15,100\n11,16,100\n11,17,100\n11,19,100\n11,20,100\n12,14,102\n12,16,102\n12,17,102\n12,18,100\n12,21,100\n12,22,102\n12,23,102\n13,17,100\n13,18,100\n13,20,102\n13,21,103\n14,17,103\n14,18,103\n14,20,100\n14,22,103\n15,18,100\n15,21,100\n15,23,102\n16,18,101\n16,19,103\n16,20,103\n16,22,105\n16,23,103\n17,19,100\n17,20,100\n17,21,100\n18,19,100\n18,20,101\n18,22,103\n19,20,103\n19,21,102\n19,22,104\n20,22,100\n20,23,100\n21,22,101\n21,23,104\n\nOh—and before you send your pick, it'd be great if you put it in a tiny JSON note like this so it's easy to read by the rest of the team:\n\n{\n  \"solution\": [<selected_kit_id>, <selected_kit_id>, ...]\n}\n\nHere \"solution\" is just the list of which kits you want in the box — put each kit's identifier in the array. This is just a sketch of the shape I expect, not your final answer; replace the placeholders with the actual identifiers from the instance.\n\nPlease use the identifiers exactly as they appear in the instance input — don't rename them or invent new labels.\n- for example: \"Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.\"",
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  {
    "task_name": "QKP",
    "problem_type": "QKP",
    "instruction": "I was loading up a mobile repair van and had to pick which tools to bring so the total weight fits in the back. Each tool can help on its own with a certain service value, and some tool pairs make jobs faster or smoother — adding a little extra value if both are on board. The trick is to pick a combination that gives the highest possible total service value (add up every chosen tool’s individual value, and include any extra bonuses for pairs that are both included) while making sure the summed weight doesn’t exceed the van’s storage limit; every tool is unique, so only one of each can be taken. The concrete weights, values, and pair bonuses are shown below.\n\n# total_tools_count=25\n# van_weight_capacity=12157\n# tool_identifiers=1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25\n\ntool_id,tool_service_value\n1,1377\n2,805\n3,502\n4,661\n5,603\n6,714\n7,722\n8,1782\n9,1162\n10,1524\n11,1835\n12,1097\n13,526\n14,1845\n15,1315\n16,1688\n17,1057\n18,400\n19,1622\n20,980\n21,873\n22,1842\n23,903\n24,460\n25,634\n\ntool_id,tool_weight\n1,1207\n2,635\n3,332\n4,491\n5,433\n6,544\n7,552\n8,1612\n9,992\n10,1354\n11,1665\n12,927\n13,356\n14,1675\n15,1145\n16,1518\n17,887\n18,230\n19,1452\n20,810\n21,703\n22,1672\n23,733\n24,290\n25,464\n\ntool_i_id,tool_j_id,pair_bonus_service\n1,2,483\n1,3,1091\n1,4,835\n1,5,1327\n1,6,1394\n1,8,1051\n1,9,583\n1,10,1026\n1,11,833\n1,12,1423\n1,13,696\n1,14,1188\n1,15,539\n1,17,614\n1,19,480\n1,20,1405\n1,21,987\n1,22,378\n1,24,612\n1,25,827\n2,4,343\n2,5,1031\n2,6,830\n2,7,793\n2,8,393\n2,10,693\n2,11,824\n2,12,505\n2,13,723\n2,14,960\n2,15,387\n2,16,654\n2,17,266\n2,18,845\n2,19,1574\n2,23,1246\n3,5,256\n3,6,743\n3,8,437\n3,9,347\n3,11,604\n3,12,727\n3,13,548\n3,15,957\n3,16,360\n3,17,941\n3,19,615\n3,20,317\n3,21,935\n3,23,767\n3,25,824\n4,7,537\n4,11,906\n4,12,242\n4,16,543\n4,17,952\n4,18,1591\n4,19,1531\n4,20,1155\n4,22,1496\n4,23,462\n4,25,753\n5,6,394\n5,7,630\n5,8,704\n5,9,664\n5,10,672\n5,11,335\n5,12,652\n5,13,619\n5,15,705\n5,18,989\n5,19,1134\n5,21,350\n5,23,1226\n5,24,469\n5,25,1190\n6,7,747\n6,8,955\n6,10,862\n6,11,930\n6,12,596\n6,13,845\n6,15,512\n6,16,702\n6,17,804\n6,18,643\n6,19,539\n6,21,753\n6,22,387\n6,23,515\n6,24,516\n6,25,648\n7,8,786\n7,10,679\n7,11,736\n7,12,755\n7,13,815\n7,15,799\n7,16,1120\n7,19,1009\n7,20,1118\n7,21,1086\n7,22,460\n7,23,664\n7,24,577\n8,9,1521\n8,10,608\n8,11,1393\n8,13,1005\n8,14,1018\n8,15,613\n8,16,1062\n8,17,927\n8,18,1203\n8,20,794\n8,22,1293\n8,23,1018\n8,24,1280\n8,25,1581\n9,11,557\n9,14,500\n9,15,274\n9,18,744\n9,19,588\n9,20,293\n9,21,615\n9,22,726\n9,23,526\n9,24,622\n9,25,659\n10,13,461\n10,14,251\n10,15,623\n10,16,534\n10,17,502\n10,20,238\n10,21,638\n10,22,422\n10,25,1075\n11,12,1515\n11,14,1581\n11,15,1501\n11,16,615\n11,17,601\n11,18,1620\n11,20,1073\n11,21,1025\n11,22,1527\n11,23,992\n11,25,722\n12,13,758\n12,14,459\n12,15,803\n12,16,758\n12,17,787\n12,18,573\n12,21,353\n12,22,403\n12,23,640\n12,24,570\n12,25,336\n13,14,975\n13,15,862\n13,17,579\n13,18,402\n13,20,705\n13,21,422\n13,22,421\n13,23,604\n14,15,480\n14,16,246\n14,17,659\n14,18,621\n14,22,222\n14,25,1318\n15,16,859\n15,17,1614\n15,18,753\n15,19,784\n15,20,913\n15,21,369\n15,22,1111\n15,23,465\n16,17,780\n16,18,317\n16,19,183\n16,20,488\n16,21,392\n16,24,235\n16,25,538\n17,18,401\n17,19,396\n17,20,598\n17,22,403\n17,23,214\n17,24,498\n17,25,410\n18,19,358\n18,20,413\n18,21,168\n18,22,342\n18,23,690\n18,24,583\n18,25,659\n19,20,583\n19,21,647\n19,22,440\n19,23,1046\n19,24,549\n19,25,681\n20,21,767\n20,22,586\n20,25,559\n21,22,428\n21,23,688\n\nYou can just show your pick as a tiny JSON snippet so it's easy to read and copy back — something like this:\n\n{\n  \"solution\": [<selected_tool_id>, <selected_tool_id>, ...]\n}\n\n\"solution\" is the list of tool IDs you'd bring in the van; each <selected_tool_id> is a placeholder for one of the tools from the instance. Think of it like filling in a simple checklist: put the exact tool IDs from the problem inside the brackets. This JSON is only a sketch of the shape I expect, not the final selection.\n\nPlease make sure to use the identifiers exactly as they appear in the instance input — don't rename them or invent new labels.\n\n- for example: \"Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.\"",
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    "problem_type": "QKP",
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The full breakdown of items, sizes, values and pair bonuses is shown below.\n\nThere are 20 distinct accessories (0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19) and the display case can hold up to 5093 units of space.\nAccessory 0 has a base sales value of 2222.\nAccessory 1 has a base sales value of 1173.\nAccessory 2 has a base sales value of 1910.\nAccessory 3 has a base sales value of 2867.\nAccessory 4 has a base sales value of 2945.\nAccessory 5 has a base sales value of 508.\nAccessory 6 has a base sales value of 437.\nAccessory 7 has a base sales value of 2582.\nAccessory 8 has a base sales value of 2044.\nAccessory 9 has a base sales value of 2464.\nAccessory 10 has a base sales value of 140.\nAccessory 11 has a base sales value of 1914.\nAccessory 12 has a base sales value of 1860.\nAccessory 13 has a base sales value of 2346.\nAccessory 14 has a base sales value of 1091.\nAccessory 15 has a base sales value of 774.\nAccessory 16 has a base sales value of 2590.\nAccessory 17 has a base sales value of 2325.\nAccessory 18 has a base sales value of 2748.\nAccessory 19 has a base sales value of 1030.\nAccessory 0 requires 2584 units of display space.\nAccessory 1 requires 260 units of display space.\nAccessory 2 requires 1664 units of display space.\nAccessory 3 requires 336 units of display space.\nAccessory 4 requires 2866 units of display space.\nAccessory 5 requires 1510 units of display space.\nAccessory 6 requires 930 units of display space.\nAccessory 7 requires 2125 units of display space.\nAccessory 8 requires 2277 units of display space.\nAccessory 9 requires 1214 units of display space.\nAccessory 10 requires 2879 units of display space.\nAccessory 11 requires 675 units of display space.\nAccessory 12 requires 1266 units of display space.\nAccessory 13 requires 925 units of display space.\nAccessory 14 requires 1783 units of display space.\nAccessory 15 requires 2200 units of display space.\nAccessory 16 requires 309 units of display space.\nAccessory 17 requires 2200 units of display space.\nAccessory 18 requires 802 units of display space.\nAccessory 19 requires 2657 units of display space.\nAccessories 0 and 4 together yield a cross-selling bonus of 2692.\nAccessories 0 and 5 together yield a cross-selling bonus of 714.\nAccessories 0 and 12 together yield a cross-selling bonus of 2943.\nAccessories 0 and 13 together yield a cross-selling bonus of 537.\nAccessories 0 and 16 together yield a cross-selling bonus of 2430.\nAccessories 1 and 4 together yield a cross-selling bonus of 1188.\nAccessories 1 and 10 together yield a cross-selling bonus of 3007.\nAccessories 1 and 11 together yield a cross-selling bonus of 1158.\nAccessories 1 and 14 together yield a cross-selling bonus of 3046.\nAccessories 2 and 13 together yield a cross-selling bonus of 1659.\nAccessories 2 and 19 together yield a cross-selling bonus of 2834.\nAccessories 3 and 4 together yield a cross-selling bonus of 818.\nAccessories 3 and 8 together yield a cross-selling bonus of 2519.\nAccessories 3 and 11 together yield a cross-selling bonus of 851.\nAccessories 3 and 14 together yield a cross-selling bonus of 1504.\nAccessories 4 and 10 together yield a cross-selling bonus of 2437.\nAccessories 4 and 11 together yield a cross-selling bonus of 2957.\nAccessories 4 and 13 together yield a cross-selling bonus of 655.\nAccessories 4 and 14 together yield a cross-selling bonus of 2657.\nAccessories 4 and 15 together yield a cross-selling bonus of 1520.\nAccessories 4 and 17 together yield a cross-selling bonus of 2964.\nAccessories 4 and 18 together yield a cross-selling bonus of 850.\nAccessories 5 and 8 together yield a cross-selling bonus of 2681.\nAccessories 5 and 10 together yield a cross-selling bonus of 141.\nAccessories 5 and 11 together yield a cross-selling bonus of 3196.\nAccessories 5 and 14 together yield a cross-selling bonus of 1571.\nAccessories 5 and 19 together yield a cross-selling bonus of 2933.\nAccessories 6 and 16 together yield a cross-selling bonus of 3093.\nAccessories 6 and 18 together yield a cross-selling bonus of 3002.\nAccessories 7 and 17 together yield a cross-selling bonus of 1558.\nAccessories 7 and 18 together yield a cross-selling bonus of 241.\nAccessories 8 and 9 together yield a cross-selling bonus of 1952.\nAccessories 8 and 11 together yield a cross-selling bonus of 966.\nAccessories 8 and 17 together yield a cross-selling bonus of 2151.\nAccessories 8 and 18 together yield a cross-selling bonus of 898.\nAccessories 9 and 15 together yield a cross-selling bonus of 576.\nAccessories 9 and 18 together yield a cross-selling bonus of 156.\nAccessories 10 and 15 together yield a cross-selling bonus of 984.\nAccessories 10 and 18 together yield a cross-selling bonus of 3038.\nAccessories 11 and 12 together yield a cross-selling bonus of 1238.\nAccessories 11 and 13 together yield a cross-selling bonus of 2079.\nAccessories 11 and 14 together yield a cross-selling bonus of 970.\nAccessories 11 and 19 together yield a cross-selling bonus of 1517.\nAccessories 12 and 13 together yield a cross-selling bonus of 2933.\nAccessories 13 and 19 together yield a cross-selling bonus of 2157.\nAccessories 14 and 15 together yield a cross-selling bonus of 792.\nAccessories 14 and 16 together yield a cross-selling bonus of 1946.\nAccessories 14 and 19 together yield a cross-selling bonus of 1436.\nA selection must respect the 5093 limit and maximize the sum of base sales values plus any cross-selling bonuses.\n\nIf you'd like the final pick in a simple, copy‑pasteable form, just send it back using this JSON layout:\n\n{\n  \"solution\": [<selected_item_id>, <selected_item_id>, ...]\n}\n\nThis \"solution\" list is where you put the accessories you choose for the display case — just drop in each selected item's identifier exactly as it appears in the instance. Think of it like a little checklist: the array holds the item labels you want to show. The JSON above is only a sketch of the shape I expect, not the actual answer.\n\nPlease use the exact identifiers from the instance input — no renaming and no new labels.\n\"Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.\"",
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    "instruction": "Someone in the kitchen is trying to build a killer tasting platter from a set of small plates. Each option has a satisfaction score and uses some of the platter’s limited space, and a few specific pairings add extra enjoyment if both dishes make the cut. The plan is to choose a selection of different plates (no repeats) that stays within the platter’s capacity, and to maximize the total satisfaction by summing the plates’ scores plus any bonus points for chosen pairs. Exact weights, scores, and combo values are shown below.\n\n{\n  \"num_plates\": 25,\n  \"platter_capacity\": 814,\n  \"plate_ids\": [\n    0,\n    1,\n    2,\n    3,\n    4,\n    5,\n    6,\n    7,\n    8,\n    9,\n    10,\n    11,\n    12,\n    13,\n    14,\n    15,\n    16,\n    17,\n    18,\n    19,\n    20,\n    21,\n    22,\n    23,\n    24\n  ],\n  \"linear\": [\n    {\n      \"plate_id\": 0,\n      \"satisfaction_score\": 140\n    },\n    {\n      \"plate_id\": 1,\n      \"satisfaction_score\": 148\n    },\n    {\n      \"plate_id\": 2,\n      \"satisfaction_score\": 104\n    },\n    {\n      \"plate_id\": 3,\n      \"satisfaction_score\": 147\n    },\n    {\n      \"plate_id\": 4,\n      \"satisfaction_score\": 116\n    },\n    {\n      \"plate_id\": 5,\n      \"satisfaction_score\": 123\n    },\n    {\n      \"plate_id\": 6,\n      \"satisfaction_score\": 105\n    },\n    {\n      \"plate_id\": 7,\n      \"satisfaction_score\": 101\n    },\n    {\n      \"plate_id\": 8,\n      \"satisfaction_score\": 116\n    },\n    {\n      \"plate_id\": 9,\n      \"satisfaction_score\": 123\n    },\n    {\n      \"plate_id\": 10,\n      \"satisfaction_score\": 132\n    },\n    {\n      \"plate_id\": 11,\n      \"satisfaction_score\": 117\n    },\n    {\n      \"plate_id\": 12,\n      \"satisfaction_score\": 128\n    },\n    {\n      \"plate_id\": 13,\n      \"satisfaction_score\": 140\n    },\n    {\n      \"plate_id\": 14,\n      \"satisfaction_score\": 102\n    },\n    {\n      \"plate_id\": 15,\n      \"satisfaction_score\": 115\n    },\n    {\n      \"plate_id\": 16,\n      \"satisfaction_score\": 129\n    },\n    {\n      \"plate_id\": 17,\n      \"satisfaction_score\": 109\n    },\n    {\n      \"plate_id\": 18,\n      \"satisfaction_score\": 119\n    },\n    {\n      \"plate_id\": 19,\n      \"satisfaction_score\": 111\n    },\n    {\n      \"plate_id\": 20,\n      \"satisfaction_score\": 144\n    },\n    {\n      \"plate_id\": 21,\n      \"satisfaction_score\": 131\n    },\n    {\n      \"plate_id\": 22,\n      \"satisfaction_score\": 103\n    },\n    {\n      \"plate_id\": 23,\n      \"satisfaction_score\": 106\n    },\n    {\n      \"plate_id\": 24,\n      \"satisfaction_score\": 119\n    }\n  ],\n  \"weights\": [\n    {\n      \"plate_id\": 0,\n      \"plate_space\": 131\n    },\n    {\n      \"plate_id\": 1,\n      \"plate_space\": 137\n    },\n    {\n      \"plate_id\": 2,\n      \"plate_space\": 100\n    },\n    {\n      \"plate_id\": 3,\n      \"plate_space\": 109\n    },\n    {\n      \"plate_id\": 4,\n      \"plate_space\": 142\n    },\n    {\n      \"plate_id\": 5,\n      \"plate_space\": 123\n    },\n    {\n      \"plate_id\": 6,\n      \"plate_space\": 119\n    },\n    {\n      \"plate_id\": 7,\n      \"plate_space\": 126\n    },\n    {\n      \"plate_id\": 8,\n      \"plate_space\": 124\n    },\n    {\n      \"plate_id\": 9,\n      \"plate_space\": 128\n    },\n    {\n      \"plate_id\": 10,\n      \"plate_space\": 106\n    },\n    {\n      \"plate_id\": 11,\n      \"plate_space\": 149\n    },\n    {\n      \"plate_id\": 12,\n      \"plate_space\": 104\n    },\n    {\n      \"plate_id\": 13,\n      \"plate_space\": 119\n    },\n    {\n      \"plate_id\": 14,\n      \"plate_space\": 120\n    },\n    {\n      \"plate_id\": 15,\n      \"plate_space\": 128\n    },\n    {\n      \"plate_id\": 16,\n      \"plate_space\": 117\n    },\n    {\n      \"plate_id\": 17,\n      \"plate_space\": 133\n    },\n    {\n      \"plate_id\": 18,\n      \"plate_space\": 119\n    },\n    {\n      \"plate_id\": 19,\n      \"plate_space\": 102\n    },\n    {\n      \"plate_id\": 20,\n      \"plate_space\": 146\n    },\n    {\n      \"plate_id\": 21,\n      \"plate_space\": 140\n    },\n    {\n      \"plate_id\": 22,\n      \"plate_space\": 104\n    },\n    {\n      \"plate_id\": 23,\n      \"plate_space\": 117\n    },\n    {\n      \"plate_id\": 24,\n      \"plate_space\": 137\n    }\n  ],\n  \"quadratic\": [\n    {\n      \"plate_i_id\": 0,\n      \"plate_j_id\": 1,\n      \"combo_bonus\": 132\n    },\n    {\n      \"plate_i_id\": 0,\n      \"plate_j_id\": 4,\n      \"combo_bonus\": 148\n    },\n    {\n      \"plate_i_id\": 0,\n      \"plate_j_id\": 5,\n      \"combo_bonus\": 129\n    },\n    {\n      \"plate_i_id\": 0,\n      \"plate_j_id\": 6,\n      \"combo_bonus\": 126\n    },\n    {\n      \"plate_i_id\": 0,\n      \"plate_j_id\": 7,\n      \"combo_bonus\": 110\n    },\n    {\n      \"plate_i_id\": 0,\n      \"plate_j_id\": 8,\n      \"combo_bonus\": 102\n    },\n    {\n      \"plate_i_id\": 0,\n      \"plate_j_id\": 11,\n      \"combo_bonus\": 121\n    },\n    {\n      \"plate_i_id\": 0,\n      \"plate_j_id\": 12,\n      \"combo_bonus\": 114\n    },\n    {\n      \"plate_i_id\": 0,\n      \"plate_j_id\": 14,\n      \"combo_bonus\": 126\n    },\n    {\n      \"plate_i_id\": 0,\n      \"plate_j_id\": 15,\n      \"combo_bonus\": 104\n    },\n    {\n      \"plate_i_id\": 0,\n      \"plate_j_id\": 17,\n      \"combo_bonus\": 137\n    },\n    {\n      \"plate_i_id\": 0,\n      \"plate_j_id\": 18,\n      \"combo_bonus\": 140\n    },\n    {\n      \"plate_i_id\": 0,\n      \"plate_j_id\": 19,\n      \"combo_bonus\": 125\n    },\n    {\n      \"plate_i_id\": 0,\n      \"plate_j_id\": 20,\n      \"combo_bonus\": 111\n    },\n    {\n      \"plate_i_id\": 0,\n      \"plate_j_id\": 21,\n      \"combo_bonus\": 123\n    },\n    {\n      \"plate_i_id\": 0,\n      \"plate_j_id\": 22,\n      \"combo_bonus\": 139\n    },\n    {\n      \"plate_i_id\": 0,\n      \"plate_j_id\": 24,\n      \"combo_bonus\": 116\n    },\n    {\n      \"plate_i_id\": 1,\n      \"plate_j_id\": 3,\n      \"combo_bonus\": 106\n    },\n    {\n      \"plate_i_id\": 1,\n      \"plate_j_id\": 4,\n      \"combo_bonus\": 109\n    },\n    {\n      \"plate_i_id\": 1,\n      \"plate_j_id\": 5,\n      \"combo_bonus\": 131\n    },\n    {\n      \"plate_i_id\": 1,\n      \"plate_j_id\": 6,\n      \"combo_bonus\": 146\n    },\n    {\n      \"plate_i_id\": 1,\n      \"plate_j_id\": 7,\n      \"combo_bonus\": 139\n    },\n    {\n      \"plate_i_id\": 1,\n      \"plate_j_id\": 8,\n      \"combo_bonus\": 104\n    },\n    {\n      \"plate_i_id\": 1,\n      \"plate_j_id\": 9,\n      \"combo_bonus\": 103\n    },\n    {\n      \"plate_i_id\": 1,\n      \"plate_j_id\": 10,\n      \"combo_bonus\": 111\n    },\n    {\n      \"plate_i_id\": 1,\n      \"plate_j_id\": 12,\n      \"combo_bonus\": 127\n    },\n    {\n      \"plate_i_id\": 1,\n      \"plate_j_id\": 13,\n      \"combo_bonus\": 117\n    },\n    {\n      \"plate_i_id\": 1,\n      \"plate_j_id\": 16,\n      \"combo_bonus\": 140\n    },\n    {\n      \"plate_i_id\": 1,\n      \"plate_j_id\": 17,\n      \"combo_bonus\": 110\n    },\n    {\n      \"plate_i_id\": 1,\n      \"plate_j_id\": 18,\n      \"combo_bonus\": 123\n    },\n    {\n      \"plate_i_id\": 1,\n      \"plate_j_id\": 19,\n      \"combo_bonus\": 110\n    },\n    {\n      \"plate_i_id\": 1,\n      \"plate_j_id\": 20,\n      \"combo_bonus\": 126\n    },\n    {\n      \"plate_i_id\": 1,\n      \"plate_j_id\": 21,\n      \"combo_bonus\": 130\n    },\n    {\n      \"plate_i_id\": 1,\n      \"plate_j_id\": 22,\n      \"combo_bonus\": 112\n    },\n    {\n      \"plate_i_id\": 1,\n      \"plate_j_id\": 23,\n      \"combo_bonus\": 121\n    },\n    {\n      \"plate_i_id\": 2,\n      \"plate_j_id\": 3,\n      \"combo_bonus\": 134\n    },\n    {\n      \"plate_i_id\": 2,\n      \"plate_j_id\": 4,\n      \"combo_bonus\": 104\n    },\n    {\n      \"plate_i_id\": 2,\n      \"plate_j_id\": 6,\n      \"combo_bonus\": 136\n    },\n    {\n      \"plate_i_id\": 2,\n      \"plate_j_id\": 7,\n      \"combo_bonus\": 144\n    },\n    {\n      \"plate_i_id\": 2,\n      \"plate_j_id\": 8,\n      \"combo_bonus\": 142\n    },\n    {\n      \"plate_i_id\": 2,\n      \"plate_j_id\": 9,\n      \"combo_bonus\": 109\n    },\n    {\n      \"plate_i_id\": 2,\n      \"plate_j_id\": 10,\n      \"combo_bonus\": 110\n    },\n    {\n      \"plate_i_id\": 2,\n      \"plate_j_id\": 11,\n      \"combo_bonus\": 133\n    },\n    {\n      \"plate_i_id\": 2,\n      \"plate_j_id\": 12,\n      \"combo_bonus\": 134\n    },\n    {\n      \"plate_i_id\": 2,\n      \"plate_j_id\": 13,\n      \"combo_bonus\": 108\n    },\n    {\n      \"plate_i_id\": 2,\n      \"plate_j_id\": 14,\n      \"combo_bonus\": 115\n    },\n    {\n      \"plate_i_id\": 2,\n      \"plate_j_id\": 17,\n      \"combo_bonus\": 112\n    },\n    {\n      \"plate_i_id\": 2,\n      \"plate_j_id\": 18,\n      \"combo_bonus\": 133\n    },\n    {\n      \"plate_i_id\": 2,\n      \"plate_j_id\": 20,\n      \"combo_bonus\": 111\n    },\n    {\n      \"plate_i_id\": 2,\n      \"plate_j_id\": 21,\n      \"combo_bonus\": 123\n    },\n    {\n      \"plate_i_id\": 2,\n      \"plate_j_id\": 22,\n      \"combo_bonus\": 119\n    },\n    {\n      \"plate_i_id\": 2,\n      \"plate_j_id\": 23,\n      \"combo_bonus\": 105\n    },\n    {\n      \"plate_i_id\": 2,\n      \"plate_j_id\": 24,\n      \"combo_bonus\": 101\n    },\n    {\n      \"plate_i_id\": 3,\n      \"plate_j_id\": 4,\n      \"combo_bonus\": 123\n    },\n    {\n      \"plate_i_id\": 3,\n      \"plate_j_id\": 5,\n      \"combo_bonus\": 131\n    },\n    {\n      \"plate_i_id\": 3,\n      \"plate_j_id\": 8,\n      \"combo_bonus\": 105\n    },\n    {\n      \"plate_i_id\": 3,\n      \"plate_j_id\": 9,\n      \"combo_bonus\": 142\n    },\n    {\n      \"plate_i_id\": 3,\n      \"plate_j_id\": 10,\n      \"combo_bonus\": 131\n    },\n    {\n      \"plate_i_id\": 3,\n      \"plate_j_id\": 11,\n      \"combo_bonus\": 110\n    },\n    {\n      \"plate_i_id\": 3,\n      \"plate_j_id\": 12,\n      \"combo_bonus\": 109\n    },\n    {\n      \"plate_i_id\": 3,\n      \"plate_j_id\": 14,\n      \"combo_bonus\": 112\n    },\n    {\n      \"plate_i_id\": 3,\n      \"plate_j_id\": 15,\n      \"combo_bonus\": 101\n    },\n    {\n      \"plate_i_id\": 3,\n      \"plate_j_id\": 17,\n      \"combo_bonus\": 136\n    },\n    {\n      \"plate_i_id\": 3,\n      \"plate_j_id\": 19,\n      \"combo_bonus\": 113\n    },\n    {\n      \"plate_i_id\": 3,\n      \"plate_j_id\": 20,\n      \"combo_bonus\": 114\n    },\n    {\n      \"plate_i_id\": 3,\n      \"plate_j_id\": 22,\n      \"combo_bonus\": 127\n    },\n    {\n      \"plate_i_id\": 3,\n      \"plate_j_id\": 23,\n      \"combo_bonus\": 113\n    },\n    {\n      \"plate_i_id\": 3,\n      \"plate_j_id\": 24,\n      \"combo_bonus\": 114\n    },\n    {\n      \"plate_i_id\": 4,\n      \"plate_j_id\": 5,\n      \"combo_bonus\": 136\n    },\n    {\n      \"plate_i_id\": 4,\n      \"plate_j_id\": 6,\n      \"combo_bonus\": 137\n    },\n    {\n      \"plate_i_id\": 4,\n      \"plate_j_id\": 7,\n      \"combo_bonus\": 107\n    },\n    {\n      \"plate_i_id\": 4,\n      \"plate_j_id\": 8,\n      \"combo_bonus\": 141\n    },\n    {\n      \"plate_i_id\": 4,\n      \"plate_j_id\": 9,\n      \"combo_bonus\": 107\n    },\n    {\n      \"plate_i_id\": 4,\n      \"plate_j_id\": 10,\n      \"combo_bonus\": 142\n    },\n    {\n      \"plate_i_id\": 4,\n      \"plate_j_id\": 11,\n      \"combo_bonus\": 123\n    },\n    {\n      \"plate_i_id\": 4,\n      \"plate_j_id\": 12,\n      \"combo_bonus\": 109\n    },\n    {\n      \"plate_i_id\": 4,\n      \"plate_j_id\": 13,\n      \"combo_bonus\": 140\n    },\n    {\n      \"plate_i_id\": 4,\n      \"plate_j_id\": 15,\n      \"combo_bonus\": 111\n    },\n    {\n      \"plate_i_id\": 4,\n      \"plate_j_id\": 16,\n      \"combo_bonus\": 140\n    },\n    {\n      \"plate_i_id\": 4,\n      \"plate_j_id\": 17,\n      \"combo_bonus\": 109\n    },\n    {\n      \"plate_i_id\": 4,\n      \"plate_j_id\": 18,\n      \"combo_bonus\": 133\n    },\n    {\n      \"plate_i_id\": 4,\n      \"plate_j_id\": 19,\n      \"combo_bonus\": 133\n    },\n    {\n      \"plate_i_id\": 4,\n      \"plate_j_id\": 20,\n      \"combo_bonus\": 139\n    },\n    {\n      \"plate_i_id\": 4,\n      \"plate_j_id\": 21,\n      \"combo_bonus\": 125\n    },\n    {\n      \"plate_i_id\": 4,\n      \"plate_j_id\": 22,\n      \"combo_bonus\": 144\n    },\n    {\n      \"plate_i_id\": 4,\n      \"plate_j_id\": 23,\n      \"combo_bonus\": 103\n    },\n    {\n      \"plate_i_id\": 5,\n      \"plate_j_id\": 6,\n      \"combo_bonus\": 137\n    },\n    {\n      \"plate_i_id\": 5,\n      \"plate_j_id\": 7,\n      \"combo_bonus\": 121\n    },\n    {\n      \"plate_i_id\": 5,\n      \"plate_j_id\": 8,\n      \"combo_bonus\": 132\n    },\n    {\n      \"plate_i_id\": 5,\n      \"plate_j_id\": 9,\n      \"combo_bonus\": 141\n    },\n    {\n      \"plate_i_id\": 5,\n      \"plate_j_id\": 10,\n      \"combo_bonus\": 102\n    },\n    {\n      \"plate_i_id\": 5,\n      \"plate_j_id\": 11,\n      \"combo_bonus\": 121\n    },\n    {\n      \"plate_i_id\": 5,\n      \"plate_j_id\": 12,\n      \"combo_bonus\": 128\n    },\n    {\n      \"plate_i_id\": 5,\n      \"plate_j_id\": 13,\n      \"combo_bonus\": 149\n    },\n    {\n      \"plate_i_id\": 5,\n      \"plate_j_id\": 14,\n      \"combo_bonus\": 146\n    },\n    {\n      \"plate_i_id\": 5,\n      \"plate_j_id\": 17,\n      \"combo_bonus\": 104\n    },\n    {\n      \"plate_i_id\": 5,\n      \"plate_j_id\": 18,\n      \"combo_bonus\": 122\n    },\n    {\n      \"plate_i_id\": 5,\n      \"plate_j_id\": 19,\n      \"combo_bonus\": 131\n    },\n    {\n      \"plate_i_id\": 5,\n      \"plate_j_id\": 20,\n      \"combo_bonus\": 124\n    },\n    {\n      \"plate_i_id\": 5,\n      \"plate_j_id\": 23,\n      \"combo_bonus\": 122\n    },\n    {\n      \"plate_i_id\": 5,\n      \"plate_j_id\": 24,\n      \"combo_bonus\": 113\n    },\n    {\n      \"plate_i_id\": 6,\n      \"plate_j_id\": 8,\n      \"combo_bonus\": 107\n    },\n    {\n      \"plate_i_id\": 6,\n      \"plate_j_id\": 10,\n      \"combo_bonus\": 123\n    },\n    {\n      \"plate_i_id\": 6,\n      \"plate_j_id\": 12,\n      \"combo_bonus\": 124\n    },\n    {\n      \"plate_i_id\": 6,\n      \"plate_j_id\": 13,\n      \"combo_bonus\": 119\n    },\n    {\n      \"plate_i_id\": 6,\n      \"plate_j_id\": 14,\n      \"combo_bonus\": 120\n    },\n    {\n      \"plate_i_id\": 6,\n      \"plate_j_id\": 15,\n      \"combo_bonus\": 139\n    },\n    {\n      \"plate_i_id\": 6,\n      \"plate_j_id\": 16,\n      \"combo_bonus\": 139\n    },\n    {\n      \"plate_i_id\": 6,\n      \"plate_j_id\": 18,\n      \"combo_bonus\": 101\n    },\n    {\n      \"plate_i_id\": 6,\n      \"plate_j_id\": 19,\n      \"combo_bonus\": 149\n    },\n    {\n      \"plate_i_id\": 6,\n      \"plate_j_id\": 20,\n      \"combo_bonus\": 137\n    },\n    {\n      \"plate_i_id\": 6,\n      \"plate_j_id\": 21,\n      \"combo_bonus\": 142\n    },\n    {\n      \"plate_i_id\": 6,\n      \"plate_j_id\": 23,\n      \"combo_bonus\": 129\n    },\n    {\n      \"plate_i_id\": 7,\n      \"plate_j_id\": 8,\n      \"combo_bonus\": 127\n    },\n    {\n      \"plate_i_id\": 7,\n      \"plate_j_id\": 9,\n      \"combo_bonus\": 125\n    },\n    {\n      \"plate_i_id\": 7,\n      \"plate_j_id\": 11,\n      \"combo_bonus\": 131\n    },\n    {\n      \"plate_i_id\": 7,\n      \"plate_j_id\": 12,\n      \"combo_bonus\": 103\n    },\n    {\n      \"plate_i_id\": 7,\n      \"plate_j_id\": 13,\n      \"combo_bonus\": 104\n    },\n    {\n      \"plate_i_id\": 7,\n      \"plate_j_id\": 14,\n      \"combo_bonus\": 149\n    },\n    {\n      \"plate_i_id\": 7,\n      \"plate_j_id\": 16,\n      \"combo_bonus\": 135\n    },\n    {\n      \"plate_i_id\": 7,\n      \"plate_j_id\": 17,\n      \"combo_bonus\": 107\n    },\n    {\n      \"plate_i_id\": 7,\n      \"plate_j_id\": 18,\n      \"combo_bonus\": 117\n    },\n    {\n      \"plate_i_id\": 7,\n      \"plate_j_id\": 19,\n      \"combo_bonus\": 144\n    },\n    {\n      \"plate_i_id\": 7,\n      \"plate_j_id\": 20,\n      \"combo_bonus\": 145\n    },\n    {\n      \"plate_i_id\": 7,\n      \"plate_j_id\": 21,\n      \"combo_bonus\": 125\n    },\n    {\n      \"plate_i_id\": 7,\n      \"plate_j_id\": 22,\n      \"combo_bonus\": 132\n    },\n    {\n      \"plate_i_id\": 7,\n      \"plate_j_id\": 23,\n      \"combo_bonus\": 102\n    },\n    {\n      \"plate_i_id\": 7,\n      \"plate_j_id\": 24,\n      \"combo_bonus\": 116\n    },\n    {\n      \"plate_i_id\": 8,\n      \"plate_j_id\": 9,\n      \"combo_bonus\": 129\n    },\n    {\n      \"plate_i_id\": 8,\n      \"plate_j_id\": 10,\n      \"combo_bonus\": 140\n    },\n    {\n      \"plate_i_id\": 8,\n      \"plate_j_id\": 11,\n      \"combo_bonus\": 129\n    },\n    {\n      \"plate_i_id\": 8,\n      \"plate_j_id\": 12,\n      \"combo_bonus\": 124\n    },\n    {\n      \"plate_i_id\": 8,\n      \"plate_j_id\": 14,\n      \"combo_bonus\": 146\n    },\n    {\n      \"plate_i_id\": 8,\n      \"plate_j_id\": 16,\n      \"combo_bonus\": 146\n    },\n    {\n      \"plate_i_id\": 8,\n      \"plate_j_id\": 17,\n      \"combo_bonus\": 140\n    },\n    {\n      \"plate_i_id\": 8,\n      \"plate_j_id\": 18,\n      \"combo_bonus\": 100\n    },\n    {\n      \"plate_i_id\": 8,\n      \"plate_j_id\": 20,\n      \"combo_bonus\": 140\n    },\n    {\n      \"plate_i_id\": 8,\n      \"plate_j_id\": 21,\n      \"combo_bonus\": 137\n    },\n    {\n      \"plate_i_id\": 8,\n      \"plate_j_id\": 22,\n      \"combo_bonus\": 114\n    },\n    {\n      \"plate_i_id\": 8,\n      \"plate_j_id\": 23,\n      \"combo_bonus\": 124\n    },\n    {\n      \"plate_i_id\": 8,\n      \"plate_j_id\": 24,\n      \"combo_bonus\": 104\n    },\n    {\n      \"plate_i_id\": 9,\n      \"plate_j_id\": 12,\n      \"combo_bonus\": 134\n    },\n    {\n      \"plate_i_id\": 9,\n      \"plate_j_id\": 13,\n      \"combo_bonus\": 119\n    },\n    {\n      \"plate_i_id\": 9,\n      \"plate_j_id\": 14,\n      \"combo_bonus\": 116\n    },\n    {\n      \"plate_i_id\": 9,\n      \"plate_j_id\": 15,\n      \"combo_bonus\": 139\n    },\n    {\n      \"plate_i_id\": 9,\n      \"plate_j_id\": 16,\n      \"combo_bonus\": 125\n    },\n    {\n      \"plate_i_id\": 9,\n      \"plate_j_id\": 17,\n      \"combo_bonus\": 125\n    },\n    {\n      \"plate_i_id\": 9,\n      \"plate_j_id\": 19,\n      \"combo_bonus\": 115\n    },\n    {\n      \"plate_i_id\": 9,\n      \"plate_j_id\": 20,\n      \"combo_bonus\": 146\n    },\n    {\n      \"plate_i_id\": 9,\n      \"plate_j_id\": 21,\n      \"combo_bonus\": 117\n    },\n    {\n      \"plate_i_id\": 9,\n      \"plate_j_id\": 23,\n      \"combo_bonus\": 132\n    },\n    {\n      \"plate_i_id\": 9,\n      \"plate_j_id\": 24,\n      \"combo_bonus\": 138\n    },\n    {\n      \"plate_i_id\": 10,\n      \"plate_j_id\": 11,\n      \"combo_bonus\": 123\n    },\n    {\n      \"plate_i_id\": 10,\n      \"plate_j_id\": 12,\n      \"combo_bonus\": 109\n    },\n    {\n      \"plate_i_id\": 10,\n      \"plate_j_id\": 13,\n      \"combo_bonus\": 100\n    },\n    {\n      \"plate_i_id\": 10,\n      \"plate_j_id\": 15,\n      \"combo_bonus\": 100\n    },\n    {\n      \"plate_i_id\": 10,\n      \"plate_j_id\": 17,\n      \"combo_bonus\": 144\n    },\n    {\n      \"plate_i_id\": 10,\n      \"plate_j_id\": 18,\n      \"combo_bonus\": 149\n    },\n    {\n      \"plate_i_id\": 10,\n      \"plate_j_id\": 19,\n      \"combo_bonus\": 144\n    },\n    {\n      \"plate_i_id\": 10,\n      \"plate_j_id\": 20,\n      \"combo_bonus\": 129\n    },\n    {\n      \"plate_i_id\": 10,\n      \"plate_j_id\": 23,\n      \"combo_bonus\": 134\n    },\n    {\n      \"plate_i_id\": 10,\n      \"plate_j_id\": 24,\n      \"combo_bonus\": 104\n    },\n    {\n      \"plate_i_id\": 11,\n      \"plate_j_id\": 12,\n      \"combo_bonus\": 149\n    },\n    {\n      \"plate_i_id\": 11,\n      \"plate_j_id\": 13,\n      \"combo_bonus\": 128\n    },\n    {\n      \"plate_i_id\": 11,\n      \"plate_j_id\": 14,\n      \"combo_bonus\": 138\n    },\n    {\n      \"plate_i_id\": 11,\n      \"plate_j_id\": 15,\n      \"combo_bonus\": 147\n    },\n    {\n      \"plate_i_id\": 11,\n 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},\n    {\n      \"plate_i_id\": 13,\n      \"plate_j_id\": 23,\n      \"combo_bonus\": 138\n    },\n    {\n      \"plate_i_id\": 14,\n      \"plate_j_id\": 15,\n      \"combo_bonus\": 142\n    },\n    {\n      \"plate_i_id\": 14,\n      \"plate_j_id\": 19,\n      \"combo_bonus\": 108\n    },\n    {\n      \"plate_i_id\": 14,\n      \"plate_j_id\": 21,\n      \"combo_bonus\": 145\n    },\n    {\n      \"plate_i_id\": 14,\n      \"plate_j_id\": 22,\n      \"combo_bonus\": 114\n    },\n    {\n      \"plate_i_id\": 14,\n      \"plate_j_id\": 23,\n      \"combo_bonus\": 109\n    },\n    {\n      \"plate_i_id\": 14,\n      \"plate_j_id\": 24,\n      \"combo_bonus\": 149\n    },\n    {\n      \"plate_i_id\": 15,\n      \"plate_j_id\": 16,\n      \"combo_bonus\": 146\n    },\n    {\n      \"plate_i_id\": 15,\n      \"plate_j_id\": 17,\n      \"combo_bonus\": 143\n    },\n    {\n      \"plate_i_id\": 15,\n      \"plate_j_id\": 18,\n      \"combo_bonus\": 124\n    },\n    {\n      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\"plate_j_id\": 18,\n      \"combo_bonus\": 134\n    },\n    {\n      \"plate_i_id\": 17,\n      \"plate_j_id\": 19,\n      \"combo_bonus\": 104\n    },\n    {\n      \"plate_i_id\": 17,\n      \"plate_j_id\": 21,\n      \"combo_bonus\": 141\n    },\n    {\n      \"plate_i_id\": 17,\n      \"plate_j_id\": 22,\n      \"combo_bonus\": 140\n    },\n    {\n      \"plate_i_id\": 18,\n      \"plate_j_id\": 19,\n      \"combo_bonus\": 105\n    },\n    {\n      \"plate_i_id\": 18,\n      \"plate_j_id\": 20,\n      \"combo_bonus\": 130\n    },\n    {\n      \"plate_i_id\": 18,\n      \"plate_j_id\": 21,\n      \"combo_bonus\": 123\n    },\n    {\n      \"plate_i_id\": 18,\n      \"plate_j_id\": 22,\n      \"combo_bonus\": 113\n    },\n    {\n      \"plate_i_id\": 18,\n      \"plate_j_id\": 23,\n      \"combo_bonus\": 135\n    },\n    {\n      \"plate_i_id\": 18,\n      \"plate_j_id\": 24,\n      \"combo_bonus\": 117\n    },\n    {\n      \"plate_i_id\": 19,\n      \"plate_j_id\": 22,\n      \"combo_bonus\": 134\n    },\n    {\n      \"plate_i_id\": 19,\n      \"plate_j_id\": 23,\n      \"combo_bonus\": 134\n    },\n    {\n      \"plate_i_id\": 20,\n      \"plate_j_id\": 21,\n      \"combo_bonus\": 118\n    },\n    {\n      \"plate_i_id\": 20,\n      \"plate_j_id\": 22,\n      \"combo_bonus\": 127\n    },\n    {\n      \"plate_i_id\": 20,\n      \"plate_j_id\": 23,\n      \"combo_bonus\": 109\n    },\n    {\n      \"plate_i_id\": 21,\n      \"plate_j_id\": 22,\n      \"combo_bonus\": 113\n    },\n    {\n      \"plate_i_id\": 21,\n      \"plate_j_id\": 23,\n      \"combo_bonus\": 123\n    },\n    {\n      \"plate_i_id\": 22,\n      \"plate_j_id\": 24,\n      \"combo_bonus\": 149\n    }\n  ]\n}\n\nOh, and when you send back your chosen plates, you can just drop them into a tiny JSON snippet so it's clear and machine-friendly. Something like this works:\n\n{\n  \"solution\": [<plate_id>, <plate_id>, ...]\n}\n\nHere \"solution\" is just the list of plates you want on the tasting platter — one entry per chosen plate, using the exact plate identifiers from the instance. This block is just a little template showing the shape I expect, not the final answer itself.\n\nPlease make sure to use the identifiers exactly as they appear in the instance input — no renaming, no made-up labels.\n- \"Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.\"",
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      ]
    },
    "solution_variant": [
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    "context_index": 46,
    "input_format": "json",
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  {
    "task_name": "QKP",
    "problem_type": "QKP",
    "instruction": "I’m loading up a rental truck with a pile of furniture and trying to decide which pieces to take so the truck doesn’t get overloaded. Each item has its own resale or reuse value, and a few specific combinations — like a matching table and set of chairs — add extra value if they travel together. The idea is to pick a bunch of items whose total weight stays under the truck’s limit while getting the biggest total payout, which is just the sum of each piece’s value plus any pair bonuses for items that were both brought along; each item can only be taken once or left behind. The exact list of pieces, weights, values and the pair bonuses is shown below.\n\n{\n  \"num_pieces\": 19,\n  \"truck_weight_limit\": 5148,\n  \"piece_ids\": [\n    \"A\",\n    \"B\",\n    \"C\",\n    \"D\",\n    \"E\",\n    \"F\",\n    \"G\",\n    \"H\",\n    \"I\",\n    \"J\",\n    \"K\",\n    \"L\",\n    \"M\",\n    \"N\",\n    \"O\",\n    \"P\",\n    \"Q\",\n    \"R\",\n    \"S\"\n  ],\n  \"linear\": [\n    {\n      \"piece_id\": \"A\",\n      \"resale_value\": 170\n    },\n    {\n      \"piece_id\": \"B\",\n      \"resale_value\": 321\n    },\n    {\n      \"piece_id\": \"C\",\n      \"resale_value\": 191\n    },\n    {\n      \"piece_id\": \"D\",\n      \"resale_value\": 422\n    },\n    {\n      \"piece_id\": \"E\",\n      \"resale_value\": 482\n    },\n    {\n      \"piece_id\": \"F\",\n      \"resale_value\": 308\n    },\n    {\n      \"piece_id\": \"G\",\n      \"resale_value\": 113\n    },\n    {\n      \"piece_id\": \"H\",\n      \"resale_value\": 248\n    },\n    {\n      \"piece_id\": \"I\",\n      \"resale_value\": 402\n    },\n    {\n      \"piece_id\": \"J\",\n      \"resale_value\": 142\n    },\n    {\n      \"piece_id\": \"K\",\n      \"resale_value\": 107\n    },\n    {\n      \"piece_id\": \"L\",\n      \"resale_value\": 236\n    },\n    {\n      \"piece_id\": \"M\",\n      \"resale_value\": 211\n    },\n    {\n      \"piece_id\": \"N\",\n      \"resale_value\": 450\n    },\n    {\n      \"piece_id\": \"O\",\n      \"resale_value\": 383\n    },\n    {\n      \"piece_id\": \"P\",\n      \"resale_value\": 403\n    },\n    {\n      \"piece_id\": \"Q\",\n      \"resale_value\": 145\n    },\n    {\n      \"piece_id\": \"R\",\n      \"resale_value\": 126\n    },\n    {\n      \"piece_id\": \"S\",\n      \"resale_value\": 435\n    }\n  ],\n  \"weights\": [\n    {\n      \"piece_id\": \"A\",\n      \"piece_weight\": 438\n    },\n    {\n      \"piece_id\": \"B\",\n      \"piece_weight\": 429\n    },\n    {\n      \"piece_id\": \"C\",\n      \"piece_weight\": 212\n    },\n    {\n      \"piece_id\": \"D\",\n      \"piece_weight\": 312\n    },\n    {\n      \"piece_id\": \"E\",\n      \"piece_weight\": 147\n    },\n    {\n      \"piece_id\": \"F\",\n      \"piece_weight\": 380\n    },\n    {\n      \"piece_id\": \"G\",\n      \"piece_weight\": 153\n    },\n    {\n      \"piece_id\": \"H\",\n      \"piece_weight\": 182\n    },\n    {\n      \"piece_id\": \"I\",\n      \"piece_weight\": 153\n    },\n    {\n      \"piece_id\": \"J\",\n      \"piece_weight\": 207\n    },\n    {\n      \"piece_id\": \"K\",\n      \"piece_weight\": 174\n    },\n    {\n      \"piece_id\": \"L\",\n      \"piece_weight\": 206\n    },\n    {\n      \"piece_id\": \"M\",\n      \"piece_weight\": 360\n    },\n    {\n      \"piece_id\": \"N\",\n      \"piece_weight\": 421\n    },\n    {\n      \"piece_id\": \"O\",\n      \"piece_weight\": 332\n    },\n    {\n      \"piece_id\": \"P\",\n      \"piece_weight\": 106\n    },\n    {\n      \"piece_id\": \"Q\",\n      \"piece_weight\": 475\n    },\n    {\n      \"piece_id\": \"R\",\n      \"piece_weight\": 238\n    },\n    {\n      \"piece_id\": \"S\",\n      \"piece_weight\": 369\n    }\n  ],\n  \"quadratic\": [\n    {\n      \"piece_a_id\": \"A\",\n      \"piece_b_id\": \"B\",\n      \"pair_bonus_value\": 270\n    },\n    {\n      \"piece_a_id\": \"A\",\n      \"piece_b_id\": \"H\",\n      \"pair_bonus_value\": 333\n    },\n    {\n      \"piece_a_id\": \"A\",\n      \"piece_b_id\": \"I\",\n      \"pair_bonus_value\": 151\n    },\n    {\n      \"piece_a_id\": \"A\",\n      \"piece_b_id\": \"J\",\n      \"pair_bonus_value\": 437\n    },\n    {\n      \"piece_a_id\": \"A\",\n      \"piece_b_id\": \"K\",\n      \"pair_bonus_value\": 118\n    },\n    {\n      \"piece_a_id\": \"A\",\n      \"piece_b_id\": \"N\",\n      \"pair_bonus_value\": 432\n    },\n    {\n      \"piece_a_id\": \"A\",\n      \"piece_b_id\": \"O\",\n      \"pair_bonus_value\": 122\n    },\n    {\n      \"piece_a_id\": \"A\",\n      \"piece_b_id\": \"P\",\n      \"pair_bonus_value\": 232\n    },\n    {\n      \"piece_a_id\": \"A\",\n      \"piece_b_id\": \"Q\",\n      \"pair_bonus_value\": 397\n    },\n    {\n      \"piece_a_id\": \"A\",\n      \"piece_b_id\": \"R\",\n      \"pair_bonus_value\": 437\n    },\n    {\n      \"piece_a_id\": \"A\",\n      \"piece_b_id\": \"S\",\n      \"pair_bonus_value\": 428\n    },\n    {\n      \"piece_a_id\": \"B\",\n      \"piece_b_id\": \"D\",\n      \"pair_bonus_value\": 469\n    },\n    {\n      \"piece_a_id\": \"B\",\n      \"piece_b_id\": \"E\",\n      \"pair_bonus_value\": 380\n    },\n    {\n      \"piece_a_id\": \"B\",\n      \"piece_b_id\": \"F\",\n      \"pair_bonus_value\": 315\n    },\n    {\n      \"piece_a_id\": \"B\",\n      \"piece_b_id\": \"G\",\n      \"pair_bonus_value\": 225\n    },\n    {\n      \"piece_a_id\": \"B\",\n      \"piece_b_id\": \"H\",\n      \"pair_bonus_value\": 191\n    },\n    {\n      \"piece_a_id\": \"B\",\n      \"piece_b_id\": \"J\",\n      \"pair_bonus_value\": 444\n    },\n    {\n      \"piece_a_id\": \"B\",\n      \"piece_b_id\": \"K\",\n      \"pair_bonus_value\": 457\n    },\n    {\n      \"piece_a_id\": \"B\",\n      \"piece_b_id\": \"L\",\n      \"pair_bonus_value\": 293\n    },\n    {\n      \"piece_a_id\": \"B\",\n      \"piece_b_id\": \"M\",\n      \"pair_bonus_value\": 461\n    },\n    {\n      \"piece_a_id\": \"B\",\n      \"piece_b_id\": \"Q\",\n      \"pair_bonus_value\": 339\n    },\n    {\n      \"piece_a_id\": \"B\",\n      \"piece_b_id\": \"S\",\n      \"pair_bonus_value\": 481\n    },\n    {\n      \"piece_a_id\": \"C\",\n      \"piece_b_id\": \"D\",\n      \"pair_bonus_value\": 330\n    },\n    {\n      \"piece_a_id\": \"C\",\n      \"piece_b_id\": \"E\",\n      \"pair_bonus_value\": 223\n    },\n    {\n      \"piece_a_id\": \"C\",\n      \"piece_b_id\": \"F\",\n      \"pair_bonus_value\": 271\n    },\n    {\n      \"piece_a_id\": \"C\",\n      \"piece_b_id\": \"G\",\n      \"pair_bonus_value\": 260\n    },\n    {\n      \"piece_a_id\": \"C\",\n      \"piece_b_id\": \"I\",\n      \"pair_bonus_value\": 210\n    },\n    {\n      \"piece_a_id\": \"C\",\n      \"piece_b_id\": \"J\",\n      \"pair_bonus_value\": 363\n    },\n    {\n      \"piece_a_id\": \"C\",\n      \"piece_b_id\": \"K\",\n      \"pair_bonus_value\": 250\n    },\n    {\n      \"piece_a_id\": \"C\",\n      \"piece_b_id\": \"L\",\n      \"pair_bonus_value\": 126\n    },\n    {\n      \"piece_a_id\": \"C\",\n      \"piece_b_id\": \"M\",\n      \"pair_bonus_value\": 312\n    },\n    {\n      \"piece_a_id\": \"C\",\n      \"piece_b_id\": \"N\",\n      \"pair_bonus_value\": 498\n    },\n    {\n      \"piece_a_id\": \"C\",\n      \"piece_b_id\": \"O\",\n      \"pair_bonus_value\": 264\n    },\n    {\n      \"piece_a_id\": \"C\",\n      \"piece_b_id\": \"P\",\n      \"pair_bonus_value\": 170\n    },\n    {\n      \"piece_a_id\": \"C\",\n      \"piece_b_id\": \"Q\",\n      \"pair_bonus_value\": 440\n    },\n    {\n      \"piece_a_id\": \"C\",\n      \"piece_b_id\": \"R\",\n      \"pair_bonus_value\": 411\n    },\n    {\n      \"piece_a_id\": \"D\",\n      \"piece_b_id\": \"E\",\n      \"pair_bonus_value\": 103\n    },\n    {\n      \"piece_a_id\": \"D\",\n      \"piece_b_id\": \"F\",\n      \"pair_bonus_value\": 450\n    },\n    {\n      \"piece_a_id\": \"D\",\n      \"piece_b_id\": \"I\",\n      \"pair_bonus_value\": 379\n    },\n    {\n      \"piece_a_id\": \"D\",\n      \"piece_b_id\": \"J\",\n      \"pair_bonus_value\": 266\n    },\n    {\n      \"piece_a_id\": \"D\",\n      \"piece_b_id\": \"K\",\n      \"pair_bonus_value\": 410\n    },\n    {\n      \"piece_a_id\": \"D\",\n      \"piece_b_id\": \"L\",\n      \"pair_bonus_value\": 339\n    },\n    {\n      \"piece_a_id\": \"D\",\n      \"piece_b_id\": \"M\",\n      \"pair_bonus_value\": 210\n    },\n    {\n      \"piece_a_id\": \"D\",\n      \"piece_b_id\": \"N\",\n      \"pair_bonus_value\": 456\n    },\n    {\n      \"piece_a_id\": \"D\",\n      \"piece_b_id\": \"R\",\n      \"pair_bonus_value\": 115\n    },\n    {\n      \"piece_a_id\": \"D\",\n      \"piece_b_id\": \"S\",\n      \"pair_bonus_value\": 255\n    },\n    {\n      \"piece_a_id\": \"E\",\n      \"piece_b_id\": \"G\",\n      \"pair_bonus_value\": 380\n    },\n    {\n      \"piece_a_id\": \"E\",\n      \"piece_b_id\": \"H\",\n      \"pair_bonus_value\": 219\n    },\n    {\n      \"piece_a_id\": \"E\",\n      \"piece_b_id\": \"I\",\n      \"pair_bonus_value\": 327\n    },\n    {\n      \"piece_a_id\": \"E\",\n      \"piece_b_id\": \"J\",\n      \"pair_bonus_value\": 262\n    },\n    {\n      \"piece_a_id\": \"E\",\n      \"piece_b_id\": \"K\",\n      \"pair_bonus_value\": 427\n    },\n    {\n      \"piece_a_id\": \"E\",\n      \"piece_b_id\": \"M\",\n      \"pair_bonus_value\": 266\n    },\n    {\n      \"piece_a_id\": \"E\",\n      \"piece_b_id\": \"N\",\n      \"pair_bonus_value\": 428\n    },\n    {\n      \"piece_a_id\": \"E\",\n     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288\n    },\n    {\n      \"piece_a_id\": \"G\",\n      \"piece_b_id\": \"Q\",\n      \"pair_bonus_value\": 424\n    },\n    {\n      \"piece_a_id\": \"H\",\n      \"piece_b_id\": \"J\",\n      \"pair_bonus_value\": 177\n    },\n    {\n      \"piece_a_id\": \"H\",\n      \"piece_b_id\": \"K\",\n      \"pair_bonus_value\": 140\n    },\n    {\n      \"piece_a_id\": \"H\",\n      \"piece_b_id\": \"L\",\n      \"pair_bonus_value\": 378\n    },\n    {\n      \"piece_a_id\": \"H\",\n      \"piece_b_id\": \"M\",\n      \"pair_bonus_value\": 299\n    },\n    {\n      \"piece_a_id\": \"H\",\n      \"piece_b_id\": \"N\",\n      \"pair_bonus_value\": 441\n    },\n    {\n      \"piece_a_id\": \"H\",\n      \"piece_b_id\": \"O\",\n      \"pair_bonus_value\": 483\n    },\n    {\n      \"piece_a_id\": \"H\",\n      \"piece_b_id\": \"P\",\n      \"pair_bonus_value\": 386\n    },\n    {\n      \"piece_a_id\": \"H\",\n      \"piece_b_id\": \"R\",\n      \"pair_bonus_value\": 224\n    },\n    {\n      \"piece_a_id\": \"H\",\n      \"piece_b_id\": \"S\",\n      \"pair_bonus_value\": 389\n    },\n    {\n      \"piece_a_id\": \"I\",\n      \"piece_b_id\": \"L\",\n      \"pair_bonus_value\": 369\n    },\n    {\n      \"piece_a_id\": \"I\",\n      \"piece_b_id\": \"M\",\n      \"pair_bonus_value\": 207\n    },\n    {\n      \"piece_a_id\": \"I\",\n      \"piece_b_id\": \"N\",\n      \"pair_bonus_value\": 424\n    },\n    {\n      \"piece_a_id\": \"I\",\n      \"piece_b_id\": \"O\",\n      \"pair_bonus_value\": 323\n    },\n    {\n      \"piece_a_id\": \"I\",\n      \"piece_b_id\": \"P\",\n      \"pair_bonus_value\": 204\n    },\n    {\n      \"piece_a_id\": \"I\",\n      \"piece_b_id\": \"S\",\n      \"pair_bonus_value\": 403\n    },\n    {\n      \"piece_a_id\": \"J\",\n      \"piece_b_id\": \"K\",\n      \"pair_bonus_value\": 364\n    },\n    {\n      \"piece_a_id\": \"J\",\n      \"piece_b_id\": \"L\",\n      \"pair_bonus_value\": 244\n    },\n    {\n      \"piece_a_id\": \"J\",\n      \"piece_b_id\": \"O\",\n      \"pair_bonus_value\": 222\n    },\n    {\n      \"piece_a_id\": \"J\",\n      \"piece_b_id\": \"P\",\n      \"pair_bonus_value\": 207\n    },\n    {\n      \"piece_a_id\": \"J\",\n      \"piece_b_id\": \"Q\",\n      \"pair_bonus_value\": 413\n    },\n    {\n      \"piece_a_id\": \"J\",\n      \"piece_b_id\": \"R\",\n      \"pair_bonus_value\": 218\n    },\n    {\n      \"piece_a_id\": \"J\",\n      \"piece_b_id\": \"S\",\n      \"pair_bonus_value\": 441\n    },\n    {\n      \"piece_a_id\": \"K\",\n      \"piece_b_id\": \"L\",\n      \"pair_bonus_value\": 199\n    },\n    {\n      \"piece_a_id\": \"K\",\n      \"piece_b_id\": \"M\",\n      \"pair_bonus_value\": 316\n    },\n    {\n      \"piece_a_id\": \"K\",\n      \"piece_b_id\": \"N\",\n      \"pair_bonus_value\": 198\n    },\n    {\n      \"piece_a_id\": \"K\",\n      \"piece_b_id\": \"P\",\n      \"pair_bonus_value\": 212\n    },\n    {\n      \"piece_a_id\": \"K\",\n      \"piece_b_id\": \"R\",\n      \"pair_bonus_value\": 467\n    },\n    {\n      \"piece_a_id\": \"L\",\n      \"piece_b_id\": \"M\",\n      \"pair_bonus_value\": 240\n    },\n    {\n      \"piece_a_id\": \"L\",\n      \"piece_b_id\": \"N\",\n      \"pair_bonus_value\": 430\n    },\n    {\n      \"piece_a_id\": \"L\",\n      \"piece_b_id\": \"O\",\n      \"pair_bonus_value\": 338\n    },\n    {\n      \"piece_a_id\": \"L\",\n      \"piece_b_id\": \"R\",\n      \"pair_bonus_value\": 450\n    },\n    {\n      \"piece_a_id\": \"L\",\n      \"piece_b_id\": \"S\",\n      \"pair_bonus_value\": 310\n    },\n    {\n      \"piece_a_id\": \"M\",\n      \"piece_b_id\": \"O\",\n      \"pair_bonus_value\": 482\n    },\n    {\n      \"piece_a_id\": \"M\",\n      \"piece_b_id\": \"P\",\n      \"pair_bonus_value\": 485\n    },\n    {\n      \"piece_a_id\": \"M\",\n      \"piece_b_id\": \"Q\",\n      \"pair_bonus_value\": 157\n    },\n    {\n      \"piece_a_id\": \"M\",\n      \"piece_b_id\": \"S\",\n      \"pair_bonus_value\": 246\n    },\n    {\n      \"piece_a_id\": \"N\",\n      \"piece_b_id\": \"O\",\n      \"pair_bonus_value\": 348\n    },\n    {\n      \"piece_a_id\": \"N\",\n      \"piece_b_id\": \"P\",\n      \"pair_bonus_value\": 150\n    },\n    {\n      \"piece_a_id\": \"N\",\n      \"piece_b_id\": \"Q\",\n      \"pair_bonus_value\": 245\n    },\n    {\n      \"piece_a_id\": \"N\",\n      \"piece_b_id\": \"R\",\n      \"pair_bonus_value\": 292\n    },\n    {\n      \"piece_a_id\": \"N\",\n      \"piece_b_id\": \"S\",\n      \"pair_bonus_value\": 445\n    },\n    {\n      \"piece_a_id\": \"O\",\n      \"piece_b_id\": \"P\",\n      \"pair_bonus_value\": 205\n    },\n    {\n      \"piece_a_id\": \"O\",\n      \"piece_b_id\": \"Q\",\n      \"pair_bonus_value\": 327\n    },\n    {\n      \"piece_a_id\": \"O\",\n      \"piece_b_id\": \"R\",\n      \"pair_bonus_value\": 489\n    },\n    {\n      \"piece_a_id\": \"O\",\n      \"piece_b_id\": \"S\",\n      \"pair_bonus_value\": 484\n    },\n    {\n      \"piece_a_id\": \"P\",\n      \"piece_b_id\": \"Q\",\n      \"pair_bonus_value\": 443\n    },\n    {\n      \"piece_a_id\": \"P\",\n      \"piece_b_id\": \"R\",\n      \"pair_bonus_value\": 196\n    },\n    {\n      \"piece_a_id\": \"P\",\n      \"piece_b_id\": \"S\",\n      \"pair_bonus_value\": 164\n    }\n  ]\n}\n\nOh, and when you send your pick, please use this simple JSON layout so it's easy to read and parse:\n\n{\n  \"solution\": [<selected_item_id>, <selected_item_id>, ...]\n}\n\nThink of \"solution\" as just a short list of the pieces you’re loading into the truck — each entry is the identifier for one item you want to take. This JSON is just a sketch of the shape I expect, not the actual answer itself.\n\nPlease use the exact identifiers from the instance input — no renaming, no new labels. \nFor example: \"Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.\"",
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        383,
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      "quadratic_coeffs": [
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      "weights": [
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      "capacity": 5148,
      "solution": [
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      "obj": 40085.0,
      "problem_type": "QKP"
    },
    "solution": [
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    "obj": 40085.0,
    "instance_variant": {
      "problem_type": "QKP",
      "num_items": 19,
      "capacity": 5148,
      "items": [
        "A",
        "B",
        "C",
        "D",
        "E",
        "F",
        "G",
        "H",
        "I",
        "J",
        "K",
        "L",
        "M",
        "N",
        "O",
        "P",
        "Q",
        "R",
        "S"
      ],
      "linear_pairs": [
        {
          "item_id": "A",
          "linear_profit": 170
        },
        {
          "item_id": "B",
          "linear_profit": 321
        },
        {
          "item_id": "C",
          "linear_profit": 191
        },
        {
          "item_id": "D",
          "linear_profit": 422
        },
        {
          "item_id": "E",
          "linear_profit": 482
        },
        {
          "item_id": "F",
          "linear_profit": 308
        },
        {
          "item_id": "G",
          "linear_profit": 113
        },
        {
          "item_id": "H",
          "linear_profit": 248
        },
        {
          "item_id": "I",
          "linear_profit": 402
        },
        {
          "item_id": "J",
          "linear_profit": 142
        },
        {
          "item_id": "K",
          "linear_profit": 107
        },
        {
          "item_id": "L",
          "linear_profit": 236
        },
        {
          "item_id": "M",
          "linear_profit": 211
        },
        {
          "item_id": "N",
          "linear_profit": 450
        },
        {
          "item_id": "O",
          "linear_profit": 383
        },
        {
          "item_id": "P",
          "linear_profit": 403
        },
        {
          "item_id": "Q",
          "linear_profit": 145
        },
        {
          "item_id": "R",
          "linear_profit": 126
        },
        {
          "item_id": "S",
          "linear_profit": 435
        }
      ],
      "weight_pairs": [
        {
          "item_id": "A",
          "weight": 438
        },
        {
          "item_id": "B",
          "weight": 429
        },
        {
          "item_id": "C",
          "weight": 212
        },
        {
          "item_id": "D",
          "weight": 312
        },
        {
          "item_id": "E",
          "weight": 147
        },
        {
          "item_id": "F",
          "weight": 380
        },
        {
          "item_id": "G",
          "weight": 153
        },
        {
          "item_id": "H",
          "weight": 182
        },
        {
          "item_id": "I",
          "weight": 153
        },
        {
          "item_id": "J",
          "weight": 207
        },
        {
          "item_id": "K",
          "weight": 174
        },
        {
          "item_id": "L",
          "weight": 206
        },
        {
          "item_id": "M",
          "weight": 360
        },
        {
          "item_id": "N",
          "weight": 421
        },
        {
          "item_id": "O",
          "weight": 332
        },
        {
          "item_id": "P",
          "weight": 106
        },
        {
          "item_id": "Q",
          "weight": 475
        },
        {
          "item_id": "R",
          "weight": 238
        },
        {
          "item_id": "S",
          "weight": 369
        }
      ],
      "quadratic_pairs": [
        {
          "item_i_id": "A",
          "item_j_id": "A",
          "quadratic_profit": 393
        },
        {
          "item_i_id": "A",
          "item_j_id": "B",
          "quadratic_profit": 270
        },
        {
          "item_i_id": "A",
          "item_j_id": "H",
          "quadratic_profit": 333
        },
        {
          "item_i_id": "A",
          "item_j_id": "I",
          "quadratic_profit": 151
        },
        {
          "item_i_id": "A",
          "item_j_id": "J",
          "quadratic_profit": 437
        },
        {
          "item_i_id": "A",
          "item_j_id": "K",
          "quadratic_profit": 118
        },
        {
          "item_i_id": "A",
          "item_j_id": "N",
          "quadratic_profit": 432
        },
        {
          "item_i_id": "A",
          "item_j_id": "O",
          "quadratic_profit": 122
        },
        {
          "item_i_id": "A",
          "item_j_id": "P",
          "quadratic_profit": 232
        },
        {
          "item_i_id": "A",
          "item_j_id": "Q",
          "quadratic_profit": 397
        },
        {
          "item_i_id": "A",
          "item_j_id": "R",
          "quadratic_profit": 437
        },
        {
          "item_i_id": "A",
          "item_j_id": "S",
          "quadratic_profit": 428
        },
        {
          "item_i_id": "B",
          "item_j_id": "B",
          "quadratic_profit": 109
        },
        {
          "item_i_id": "B",
          "item_j_id": "D",
          "quadratic_profit": 469
        },
        {
          "item_i_id": "B",
          "item_j_id": "E",
          "quadratic_profit": 380
        },
        {
          "item_i_id": "B",
          "item_j_id": "F",
          "quadratic_profit": 315
        },
        {
          "item_i_id": "B",
          "item_j_id": "G",
          "quadratic_profit": 225
        },
        {
          "item_i_id": "B",
          "item_j_id": "H",
          "quadratic_profit": 191
        },
        {
          "item_i_id": "B",
          "item_j_id": "J",
          "quadratic_profit": 444
        },
        {
          "item_i_id": "B",
          "item_j_id": "K",
          "quadratic_profit": 457
        },
        {
          "item_i_id": "B",
          "item_j_id": "L",
          "quadratic_profit": 293
        },
        {
          "item_i_id": "B",
          "item_j_id": "M",
          "quadratic_profit": 461
        },
        {
          "item_i_id": "B",
          "item_j_id": "Q",
          "quadratic_profit": 339
        },
        {
          "item_i_id": "B",
          "item_j_id": "S",
          "quadratic_profit": 481
        },
        {
          "item_i_id": "C",
          "item_j_id": "D",
          "quadratic_profit": 330
        },
        {
          "item_i_id": "C",
          "item_j_id": "E",
          "quadratic_profit": 223
        },
        {
          "item_i_id": "C",
          "item_j_id": "F",
          "quadratic_profit": 271
        },
        {
          "item_i_id": "C",
          "item_j_id": "G",
          "quadratic_profit": 260
        },
        {
          "item_i_id": "C",
          "item_j_id": "I",
          "quadratic_profit": 210
        },
        {
          "item_i_id": "C",
          "item_j_id": "J",
          "quadratic_profit": 363
        },
        {
          "item_i_id": "C",
          "item_j_id": "K",
          "quadratic_profit": 250
        },
        {
          "item_i_id": "C",
          "item_j_id": "L",
          "quadratic_profit": 126
        },
        {
          "item_i_id": "C",
          "item_j_id": "M",
          "quadratic_profit": 312
        },
        {
          "item_i_id": "C",
          "item_j_id": "N",
          "quadratic_profit": 498
        },
        {
          "item_i_id": "C",
          "item_j_id": "O",
          "quadratic_profit": 264
        },
        {
          "item_i_id": "C",
          "item_j_id": "P",
          "quadratic_profit": 170
        },
        {
          "item_i_id": "C",
          "item_j_id": "Q",
          "quadratic_profit": 440
        },
        {
          "item_i_id": "C",
          "item_j_id": "R",
          "quadratic_profit": 411
        },
        {
          "item_i_id": "D",
          "item_j_id": "E",
          "quadratic_profit": 103
        },
        {
          "item_i_id": "D",
          "item_j_id": "F",
          "quadratic_profit": 450
        },
        {
          "item_i_id": "D",
          "item_j_id": "I",
          "quadratic_profit": 379
        },
        {
          "item_i_id": "D",
          "item_j_id": "J",
          "quadratic_profit": 266
        },
        {
          "item_i_id": "D",
          "item_j_id": "K",
          "quadratic_profit": 410
        },
        {
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          "item_j_id": "L",
          "quadratic_profit": 339
        },
        {
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          "item_j_id": "M",
          "quadratic_profit": 210
        },
        {
          "item_i_id": "D",
          "item_j_id": "N",
          "quadratic_profit": 456
        },
        {
          "item_i_id": "D",
          "item_j_id": "R",
          "quadratic_profit": 115
        },
        {
          "item_i_id": "D",
          "item_j_id": "S",
          "quadratic_profit": 255
        },
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    "instruction": "Back at the makerspace there’s one shelf with a strict weight capacity and a bunch of cool equipment to pick from. Each item brings its own usefulness number and adds to the total weight, and certain item pairs produce an extra boost if they’re both on the shelf. The objective is to choose which individual items to put there — no duplicates, and don’t exceed the shelf’s weight — so that the overall usefulness (individual scores plus any pairwise boosts that apply) ends up as high as possible. The full list of weights, usefulness values, and pair bonuses is given below.\n\n# num_equipment_items=21\n# shelf_weight_capacity=35975\n# equipment_ids=1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21\n\nequipment_id,base_usefulness_value\n1,1652\n2,2905\n3,2860\n4,2583\n5,292\n6,1661\n7,1713\n8,2017\n9,2738\n10,490\n11,542\n12,1873\n13,1587\n14,2342\n15,148\n16,1623\n17,2180\n18,728\n19,1485\n20,2249\n21,420\n\nequipment_id,equipment_weight\n1,2160\n2,1544\n3,2734\n4,3144\n5,2472\n6,646\n7,2678\n8,2847\n9,2205\n10,968\n11,3049\n12,132\n13,1380\n14,1646\n15,1992\n16,801\n17,297\n18,2340\n19,801\n20,2948\n21,1058\n\nequipment_i_id,equipment_j_id,pair_bonus_usefulness\n1,3,1123\n1,11,2514\n1,16,1934\n2,3,1363\n2,7,1683\n2,8,1845\n2,11,1498\n2,13,1045\n2,20,1844\n2,21,2209\n3,6,2705\n3,12,1942\n3,13,1591\n4,5,3121\n4,15,1120\n4,19,1800\n4,21,1920\n5,6,945\n5,8,2346\n5,13,1837\n5,14,2054\n5,15,1313\n5,20,1836\n5,21,2021\n6,16,1054\n6,17,571\n6,21,2958\n7,13,1560\n7,15,1846\n7,19,2097\n8,11,1414\n8,13,2307\n8,17,531\n8,20,974\n9,11,1328\n9,18,1156\n9,21,1298\n10,13,1679\n10,19,545\n11,12,2663\n11,13,2671\n11,14,1317\n11,15,1543\n11,18,1755\n12,13,303\n12,16,1625\n12,17,805\n12,19,1104\n12,21,976\n13,19,540\n14,15,2372\n14,19,2560\n15,16,1983\n15,20,635\n16,17,2702\n16,19,1932\n17,21,2284\n\nOh, and when you send back your pick, just use this little JSON shape so it's easy to read and machine-friendly:\n\n{\n  \"solution\": [<selected_item_id>, <selected_item_id>, ...]\n}\n\n\"solution\" is just the list of items you want on the shelf — put each item's identifier in that array. Think of it like filling out a short form: which items go on the shelf (no duplicates), listed by their exact IDs. This snippet is only a sketch of the expected shape, not the actual answer.\n\nValid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.",
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        {
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        },
        {
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          "linear_profit": 2860
        },
        {
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          "linear_profit": 2583
        },
        {
          "item_id": 5,
          "linear_profit": 292
        },
        {
          "item_id": 6,
          "linear_profit": 1661
        },
        {
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          "linear_profit": 1713
        },
        {
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          "linear_profit": 2017
        },
        {
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          "linear_profit": 2738
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        {
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          "linear_profit": 490
        },
        {
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          "linear_profit": 542
        },
        {
          "item_id": 12,
          "linear_profit": 1873
        },
        {
          "item_id": 13,
          "linear_profit": 1587
        },
        {
          "item_id": 14,
          "linear_profit": 2342
        },
        {
          "item_id": 15,
          "linear_profit": 148
        },
        {
          "item_id": 16,
          "linear_profit": 1623
        },
        {
          "item_id": 17,
          "linear_profit": 2180
        },
        {
          "item_id": 18,
          "linear_profit": 728
        },
        {
          "item_id": 19,
          "linear_profit": 1485
        },
        {
          "item_id": 20,
          "linear_profit": 2249
        },
        {
          "item_id": 21,
          "linear_profit": 420
        }
      ],
      "weight_pairs": [
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          "item_id": 1,
          "weight": 2160
        },
        {
          "item_id": 2,
          "weight": 1544
        },
        {
          "item_id": 3,
          "weight": 2734
        },
        {
          "item_id": 4,
          "weight": 3144
        },
        {
          "item_id": 5,
          "weight": 2472
        },
        {
          "item_id": 6,
          "weight": 646
        },
        {
          "item_id": 7,
          "weight": 2678
        },
        {
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        },
        {
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        },
        {
          "item_id": 10,
          "weight": 968
        },
        {
          "item_id": 11,
          "weight": 3049
        },
        {
          "item_id": 12,
          "weight": 132
        },
        {
          "item_id": 13,
          "weight": 1380
        },
        {
          "item_id": 14,
          "weight": 1646
        },
        {
          "item_id": 15,
          "weight": 1992
        },
        {
          "item_id": 16,
          "weight": 801
        },
        {
          "item_id": 17,
          "weight": 297
        },
        {
          "item_id": 18,
          "weight": 2340
        },
        {
          "item_id": 19,
          "weight": 801
        },
        {
          "item_id": 20,
          "weight": 2948
        },
        {
          "item_id": 21,
          "weight": 1058
        }
      ],
      "quadratic_pairs": [
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          "item_j_id": 3,
          "quadratic_profit": 1123
        },
        {
          "item_i_id": 1,
          "item_j_id": 11,
          "quadratic_profit": 2514
        },
        {
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          "quadratic_profit": 1934
        },
        {
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        },
        {
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        },
        {
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        },
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        },
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        },
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        },
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        },
        {
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          "quadratic_profit": 1800
        },
        {
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          "quadratic_profit": 1920
        },
        {
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          "quadratic_profit": 1171
        },
        {
          "item_i_id": 5,
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          "quadratic_profit": 945
        },
        {
          "item_i_id": 5,
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          "quadratic_profit": 2346
        },
        {
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          "quadratic_profit": 1837
        },
        {
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          "quadratic_profit": 2054
        },
        {
          "item_i_id": 5,
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          "quadratic_profit": 1313
        },
        {
          "item_i_id": 5,
          "item_j_id": 20,
          "quadratic_profit": 1836
        },
        {
          "item_i_id": 5,
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          "quadratic_profit": 2021
        },
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          "quadratic_profit": 1054
        },
        {
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          "quadratic_profit": 571
        },
        {
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          "quadratic_profit": 2958
        },
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          "quadratic_profit": 1560
        },
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        },
        {
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          "quadratic_profit": 2097
        },
        {
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          "quadratic_profit": 1414
        },
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          "quadratic_profit": 2307
        },
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          "quadratic_profit": 531
        },
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          "quadratic_profit": 974
        },
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          "quadratic_profit": 1328
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          "quadratic_profit": 1156
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          "quadratic_profit": 1298
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          "quadratic_profit": 1679
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          "quadratic_profit": 545
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          "quadratic_profit": 2282
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          "quadratic_profit": 2663
        },
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          "quadratic_profit": 2671
        },
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          "quadratic_profit": 1317
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          "quadratic_profit": 1543
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          "quadratic_profit": 1755
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          "quadratic_profit": 303
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          "quadratic_profit": 1625
        },
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          "quadratic_profit": 805
        },
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          "item_j_id": 19,
          "quadratic_profit": 1104
        },
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          "item_j_id": 21,
          "quadratic_profit": 976
        },
        {
          "item_i_id": 13,
          "item_j_id": 13,
          "quadratic_profit": 1308
        },
        {
          "item_i_id": 13,
          "item_j_id": 19,
          "quadratic_profit": 540
        },
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          "item_j_id": 15,
          "quadratic_profit": 2372
        },
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          "item_j_id": 19,
          "quadratic_profit": 2560
        },
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          "item_i_id": 15,
          "item_j_id": 16,
          "quadratic_profit": 1983
        },
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          "item_i_id": 15,
          "item_j_id": 20,
          "quadratic_profit": 635
        },
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          "item_j_id": 17,
          "quadratic_profit": 2702
        },
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          "item_i_id": 16,
          "item_j_id": 19,
          "quadratic_profit": 1932
        },
        {
          "item_i_id": 17,
          "item_j_id": 21,
          "quadratic_profit": 2284
        }
      ]
    },
    "solution_variant": [
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      2,
      3,
      4,
      5,
      6,
      7,
      8,
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      11,
      12,
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    ],
    "context_index": 48,
    "input_format": "csv",
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  },
  {
    "task_name": "QKP",
    "problem_type": "QKP",
    "instruction": "Many people on the events team know the drill: one courier bag, a pile of flyers and kits, and a short list of choices. Every item adds some outreach value and some pairs bring extra bang when they’re combined. The goal is to end up with the highest possible outreach total — add the values of the chosen items and any extra pair bonuses — while respecting the bag’s weight limit and the rule that each piece is unique and can only be included once. The full breakdown of weights, values, and pair boosts is listed below.\n\nThere are 25 distinct pieces listed as A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V, W, X, Y, and the courier bag holds up to 1052 weight units.\n\n| material_id | outreach_value |\n|---|---|\n| A | 111 |\n| B | 113 |\n| C | 113 |\n| D | 119 |\n| E | 117 |\n| F | 112 |\n| G | 113 |\n| H | 122 |\n| I | 121 |\n| J | 122 |\n| K | 117 |\n| L | 114 |\n| M | 112 |\n| N | 122 |\n| O | 120 |\n| P | 114 |\n| Q | 115 |\n| R | 111 |\n| S | 113 |\n| T | 116 |\n| U | 112 |\n| V | 111 |\n| W | 122 |\n| X | 113 |\n| Y | 111 |\n\n| material_id | item_weight |\n|---|---|\n| A | 100 |\n| B | 102 |\n| C | 102 |\n| D | 108 |\n| E | 106 |\n| F | 101 |\n| G | 102 |\n| H | 111 |\n| I | 110 |\n| J | 111 |\n| K | 106 |\n| L | 103 |\n| M | 101 |\n| N | 111 |\n| O | 109 |\n| P | 103 |\n| Q | 104 |\n| R | 100 |\n| S | 102 |\n| T | 105 |\n| U | 101 |\n| V | 100 |\n| W | 111 |\n| X | 102 |\n| Y | 100 |\n\n| material_i_id | material_j_id | combo_bonus |\n|---|---|---|\n| A | B | 108 |\n| A | C | 111 |\n| A | D | 107 |\n| A | E | 108 |\n| A | F | 101 |\n| A | G | 110 |\n| A | H | 109 |\n| A | I | 104 |\n| A | J | 106 |\n| A | K | 102 |\n| A | L | 106 |\n| A | P | 101 |\n| A | Q | 105 |\n| A | R | 103 |\n| A | S | 100 |\n| A | T | 107 |\n| A | U | 103 |\n| A | V | 102 |\n| A | W | 109 |\n| A | X | 107 |\n| A | Y | 101 |\n| B | C | 107 |\n| B | D | 100 |\n| B | E | 110 |\n| B | F | 109 |\n| B | G | 111 |\n| B | H | 106 |\n| B | I | 109 |\n| B | J | 107 |\n| B | K | 104 |\n| B | L | 108 |\n| B | M | 102 |\n| B | N | 111 |\n| B | O | 108 |\n| B | R | 110 |\n| B | S | 108 |\n| B | T | 110 |\n| B | W | 111 |\n| B | X | 103 |\n| C | D | 106 |\n| C | E | 104 |\n| C | F | 104 |\n| C | G | 105 |\n| C | H | 102 |\n| C | I | 105 |\n| C | K | 105 |\n| C | L | 108 |\n| C | M | 106 |\n| C | N | 102 |\n| C | P | 107 |\n| C | Q | 110 |\n| C | R | 105 |\n| C | S | 110 |\n| C | T | 101 |\n| C | U | 111 |\n| C | V | 108 |\n| C | W | 111 |\n| C | X | 101 |\n| D | E | 105 |\n| D | F | 101 |\n| D | G | 109 |\n| D | H | 105 |\n| D | I | 101 |\n| D | K | 110 |\n| D | L | 100 |\n| D | M | 107 |\n| D | O | 108 |\n| D | P | 101 |\n| D | Q | 104 |\n| D | R | 107 |\n| D | T | 107 |\n| D | U | 105 |\n| D | V | 104 |\n| D | W | 102 |\n| D | Y | 102 |\n| E | F | 108 |\n| E | G | 105 |\n| E | H | 105 |\n| E | I | 105 |\n| E | J | 107 |\n| E | K | 105 |\n| E | M | 101 |\n| E | N | 110 |\n| E | O | 109 |\n| E | P | 108 |\n| E | Q | 101 |\n| E | X | 100 |\n| F | G | 107 |\n| F | H | 109 |\n| F | I | 104 |\n| F | J | 101 |\n| F | K | 106 |\n| F | L | 106 |\n| F | M | 105 |\n| F | N | 111 |\n| F | P | 103 |\n| F | Q | 108 |\n| F | R | 100 |\n| F | S | 109 |\n| F | T | 110 |\n| F | U | 103 |\n| F | V | 100 |\n| F | W | 107 |\n| F | X | 101 |\n| F | Y | 102 |\n| G | H | 111 |\n| G | I | 103 |\n| G | J | 106 |\n| G | M | 102 |\n| G | N | 102 |\n| G | O | 105 |\n| G | P | 104 |\n| G | Q | 100 |\n| G | R | 100 |\n| G | S | 100 |\n| G | T | 104 |\n| G | U | 107 |\n| G | W | 100 |\n| G | Y | 111 |\n| H | K | 110 |\n| H | L | 100 |\n| H | M | 109 |\n| H | O | 106 |\n| H | S | 101 |\n| H | U | 102 |\n| H | V | 111 |\n| H | X | 109 |\n| I | K | 102 |\n| I | L | 100 |\n| I | M | 104 |\n| I | N | 107 |\n| I | Q | 109 |\n| I | R | 109 |\n| I | S | 104 |\n| I | U | 102 |\n| I | W | 109 |\n| I | X | 111 |\n| J | K | 100 |\n| J | L | 100 |\n| J | M | 102 |\n| J | O | 105 |\n| J | P | 109 |\n| J | Q | 104 |\n| J | S | 106 |\n| J | T | 105 |\n| J | U | 110 |\n| J | V | 101 |\n| J | W | 110 |\n| K | L | 102 |\n| K | M | 105 |\n| K | N | 110 |\n| K | P | 100 |\n| K | Q | 104 |\n| K | R | 104 |\n| K | S | 107 |\n| K | U | 104 |\n| K | V | 110 |\n| K | X | 105 |\n| K | Y | 105 |\n| L | M | 103 |\n| L | N | 105 |\n| L | O | 103 |\n| L | P | 109 |\n| L | Q | 103 |\n| L | V | 108 |\n| L | W | 109 |\n| L | X | 101 |\n| L | Y | 100 |\n| M | N | 104 |\n| M | P | 108 |\n| M | Q | 103 |\n| M | R | 104 |\n| M | S | 102 |\n| M | V | 106 |\n| M | X | 103 |\n| M | Y | 102 |\n| N | O | 105 |\n| N | P | 109 |\n| N | S | 101 |\n| N | U | 103 |\n| N | V | 106 |\n| N | W | 102 |\n| N | X | 108 |\n| N | Y | 100 |\n| O | P | 106 |\n| O | Q | 105 |\n| O | R | 101 |\n| O | S | 105 |\n| O | T | 101 |\n| O | V | 111 |\n| O | W | 106 |\n| O | X | 110 |\n| O | Y | 100 |\n| P | Q | 111 |\n| P | U | 106 |\n| P | W | 101 |\n| P | X | 109 |\n| P | Y | 103 |\n| Q | R | 102 |\n| Q | S | 110 |\n| Q | T | 109 |\n| Q | U | 110 |\n| Q | V | 105 |\n| Q | X | 102 |\n| Q | Y | 103 |\n| R | S | 104 |\n| R | T | 111 |\n| R | V | 107 |\n| R | W | 108 |\n| R | X | 106 |\n| S | U | 100 |\n| S | V | 106 |\n| S | X | 109 |\n| T | U | 105 |\n| T | V | 108 |\n| T | W | 105 |\n| T | X | 101 |\n| T | Y | 107 |\n| U | V | 109 |\n| U | X | 111 |\n| U | Y | 110 |\n| V | W | 104 |\n| W | X | 101 |\n| W | Y | 101 |\n\nStick to the bag limit and unique-piece rule — aim for the highest outreach.\n\nOh, and when you tell me which items you’re packing, a tiny JSON snippet like this works great — super simple and keeps everything clear:\n\n{\n  \"solution\": [<selected_item_id>, <selected_item_id>, ...]\n}\n\nThink of \"solution\" as the list of item IDs you picked to put in the bag. The angle-bracket placeholders are just a sketch of the shape I expect — when you send your actual answer, swap those placeholders for the real item IDs from the instance.\n\nPlease use the item identifiers exactly as they appear in the problem input — do not rename them or invent new labels. Valid identifiers look like plain numbers such as \"1\" or \"23\", single capital letters like \"A\" or \"B\", or a capital letter followed by digits like \"A1\" or \"X7\".",
    "instance": {
      "linear_coeffs": [
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        119,
        117,
        112,
        113,
        122,
        121,
        122,
        117,
        114,
        112,
        122,
        120,
        114,
        115,
        111,
        113,
        116,
        112,
        111,
        122,
        113,
        111
      ],
      "quadratic_coeffs": [
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          108,
          101,
          110,
          109,
          104,
          106,
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        ],
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        ],
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        ],
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        ],
        [
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Think of it like filling in a shopping list: just put the exact item IDs you’re choosing inside the brackets. This JSON is just the shape I’m expecting, not your final answer.\n\nPlease use the item identifiers exactly as they appear in the instance input — don’t rename them or invent new labels. \n- for example: Valid identifiers look like plain numbers such as \"1\" or \"23\", single capital letters like \"A\" or \"B\", or a capital letter followed by digits like \"A1\" or \"X7\".",
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