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Solve the programming task below in a Python markdown code block. In a laboratory, an assistant, Nathan Wada, is measuring weight differences between sample pieces pair by pair. He is using a balance because it can more precisely measure the weight difference between two samples than a spring scale when the samples have nearly the same weight. He is occasionally asked the weight differences between pairs of samples. He can or cannot answer based on measurement results already obtained. Since he is accumulating a massive amount of measurement data, it is now not easy for him to promptly tell the weight differences. Nathan asks you to develop a program that records measurement results and automatically tells the weight differences. Input The input consists of multiple datasets. The first line of a dataset contains two integers N and M. N denotes the number of sample pieces (2 ≤ N ≤ 100,000). Each sample is assigned a unique number from 1 to N as an identifier. The rest of the dataset consists of M lines (1 ≤ M ≤ 100,000), each of which corresponds to either a measurement result or an inquiry. They are given in chronological order. A measurement result has the format, ! a b w which represents the sample piece numbered b is heavier than one numbered a by w micrograms (a ≠ b). That is, w = wb − wa, where wa and wb are the weights of a and b, respectively. Here, w is a non-negative integer not exceeding 1,000,000. You may assume that all measurements are exact and consistent. An inquiry has the format, ? a b which asks the weight difference between the sample pieces numbered a and b (a ≠ b). The last dataset is followed by a line consisting of two zeros separated by a space. Output For each inquiry, ? a b, print the weight difference in micrograms between the sample pieces numbered a and b, wb − wa, followed by a newline if the weight difference can be computed based on the measurement results prior to the inquiry. The difference can be zero, or negative as well as positive. You can assume that its absolute value is at most 1,000,000. If the difference cannot be computed based on the measurement results prior to the inquiry, print UNKNOWN followed by a newline. Example Input 2 2 ! 1 2 1 ? 1 2 2 2 ! 1 2 1 ? 2 1 4 7 ! 1 2 100 ? 2 3 ! 2 3 100 ? 2 3 ? 1 3 ! 4 3 150 ? 4 1 0 0 Output 1 -1 UNKNOWN 100 200 -50 The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Matryoshka Matryoshka is a famous Russian folk craft doll. Matryoshka can be divided into upper and lower parts, and when opened, there is another smaller doll inside. The nesting structure is such that when the small doll that appears is opened, a smaller doll is contained. <image> You found an unusually shaped matryoshka doll on your trip and bought N dolls. The shape of the i-th doll is a rectangular parallelepiped of xi × yi × zi. After watching Matryoshka for a while, you are about to put away Matryoshka. Before that, I want to reduce the space required by storing some dolls in another. When storing a doll, one other doll can be stored only in the doll that has not yet stored any doll. However, only the dolls that are stored directly are counted, and the doll that contains the doll can be stored in another doll. The stored doll becomes invisible from the outside. However, the following conditions must be met. * The doll may rotate, but each side of the rectangular parallelepiped is parallel to any side of the other rectangular parallelepiped. * After rotation, the length of the doll on the side to be stored is shorter for each of the lengths of the corresponding sides. * At most one doll can be stored directly in one doll Since the volume of the closet is limited, we want to minimize the sum of the volumes of the dolls that can be seen from the outside. Your job is to create a program that finds the minimum sum of the volumes of the dolls that are visible from the outside, which can be achieved by repeating the operation of storing the dolls any number of times. Input The input consists of multiple datasets. The maximum number of data sets does not exceed 50. Each dataset is represented in the following format. > N > x1 y1 z1 >:: >:: > xN yN zN > Each dataset consists of N + 1 rows, and the first row of the dataset is given the integer N, which represents the number of dolls. In the i-th line of the following N lines, three integers xi, yi, and zi representing the size of the i-th doll are given, separated by a half-width space. These integers satisfy 1 ≤ N, xi, yi, zi ≤ 100. > The end of the input is represented by a single zero line. > ### Output For each data set, output the minimum value of the sum of the volumes of the dolls that can be seen from the outside in one line. > ### Sample Input 2 one two Three 4 2 3 3 2 5 2 3 3 4 5 5 5 Five 1 1 1 2 2 2 3 3 3 4 4 4 5 5 5 Five 1 1 1 2 1 1 3 1 1 4 1 1 5 1 1 Ten 3 1 4 1 5 9 2 6 5 3 5 8 9 7 9 3 2 3 8 4 6 2 6 4 3 3 8 3 2 7 0 Output for Sample Input twenty four 145 125 15 864 Example Input 2 1 2 3 4 2 3 3 2 5 2 3 3 4 5 5 5 5 1 1 1 2 2 2 3 3 3 4 4 4 5 5 5 5 1 1 1 2 1 1 3 1 1 4 1 1 5 1 1 10 3 1 4 1 5 9 2 6 5 3 5 8 9 7 9 3 2 3 8 4 6 2 6 4 3 3 8 3 2 7 0 Output 24 145 125 15 864 The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Cutlet Sandwich In some worlds, there are $ X $ types of "sandwiches", $ Y $ types of "cutlets", and $ Z $ types of "curry" foods. There is a $ N $ type of "cutlet sandwich" in this world, and the $ i $ type of cutlet sandwich is made from the $ A_i $ type of sand and the $ B_i $ type of cutlet. There is also a $ M $ type of "cutlet curry", and the $ i $ type of cutlet curry is made from the $ C_i $ type of cutlet and the $ D_i $ type of curry. When you have a cutlet sandwich or cutlet curry, you can exchange it for a cutlet sandwich or cutlet curry that shares at least $ 1 $ of the ingredients. For example, if the $ a $ type sand and the $ b $ type cutlet have the raw cutlet sand, then the $ a $ type sand or any cutlet sand made from the $ b $ type cutlet. Or, you can exchange it for any cutlet curry that contains $ b $ type of cutlet. Right now, Segtree has a $ S $ kind of cutlet sandwich, but he wants to eat a $ T $ kind of cutlet curry. Determine if you can get the $ T $ kind of cutlet curry. If possible, ask for the minimum number of replacements to get the desired cutlet curry. input Input is given from standard input in the following format. $ X $ $ Y $ $ Z $ $ N $ $ M $ $ S $ $ T $ $ A_1 $ $ B_1 $ $ A_2 $ $ B_2 $ $ \ ldots $ $ A_N $ $ B_N $ $ C_1 $ $ D_1 $ $ C_2 $ $ D_2 $ $ \ ldots $ $ C_M $ $ D_M $ output Please output the minimum number of exchanges required to obtain the $ T $ type of cutlet curry. If you can't get it, print "$ -1 $" instead. However, insert a line break at the end. Constraint * $ 1 \ leq X, Y, Z, N, M \ leq 10 ^ 5 $ * $ 1 \ leq S \ leq N $ * $ 1 \ leq T \ leq M $ * $ 1 \ leq A_i \ leq X $ * $ 1 \ leq B_i \ leq Y $ * $ 1 \ leq C_i \ leq Y $ * $ 1 \ leq D_i \ leq Z $ * All inputs are integers. Input example 1 1 1 1 1 1 1 1 1 1 1 1 Output example 1 1 Input example 2 2 3 4 3 5 1 5 1 1 1 2 twenty two twenty one 3 1 3 2 3 3 3 4 Output example 2 Four Input example 3 1 2 2 1 2 1 1 1 2 1 2 twenty one Output example 3 -1 Example Input 1 1 1 1 1 1 1 1 1 1 1 Output 1 The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Constraints * 1 ≤ \|V\| ≤ 1000 * 0 ≤ \|E\| ≤ 2000 * -10000 ≤ di ≤ 10000 * There are no parallel edges * There are no self-loops Input An edge-weighted graph G (V, E) and the source r. \|V\| \|E\| r s0 t0 d0 s1 t1 d1 : s\|E\|-1 t\|E\|-1 d\|E\|-1 \|V\| is the number of vertices and \|E\| is the number of edges in G. The graph vertices are named with the numbers 0, 1,..., \|V\|-1 respectively. r is the source of the graph. si and ti represent source and target vertices of i-th edge (directed) and di represents the cost of the i-th edge. Output If the graph contains a negative cycle (a cycle whose sum of edge costs is a negative value) which is reachable from the source r, print NEGATIVE CYCLE in a line. Otherwise, print c0 c1 : c\|V\|-1 The output consists of \|V\| lines. Print the cost of the shortest path from the source r to each vertex 0, 1, ... \|V\|-1 in order. If there is no path from the source to a vertex, print "INF". Examples Input 4 5 0 0 1 2 0 2 3 1 2 -5 1 3 1 2 3 2 Output 0 2 -3 -1 Input 4 6 0 0 1 2 0 2 3 1 2 -5 1 3 1 2 3 2 3 1 0 Output NEGATIVE CYCLE Input 4 5 1 0 1 2 0 2 3 1 2 -5 1 3 1 2 3 2 Output INF 0 -5 -3 The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Initially there was an array a consisting of n integers. Positions in it are numbered from 1 to n. Exactly q queries were performed on the array. During the i-th query some segment (l_i, r_i) (1 ≤ l_i ≤ r_i ≤ n) was selected and values of elements on positions from l_i to r_i inclusive got changed to i. The order of the queries couldn't be changed and all q queries were applied. It is also known that every position from 1 to n got covered by at least one segment. We could have offered you the problem about checking if some given array (consisting of n integers with values from 1 to q) can be obtained by the aforementioned queries. However, we decided that it will come too easy for you. So the enhancement we introduced to it is the following. Some set of positions (possibly empty) in this array is selected and values of elements on these positions are set to 0. Your task is to check if this array can be obtained by the aforementioned queries. Also if it can be obtained then restore this array. If there are multiple possible arrays then print any of them. Input The first line contains two integers n and q (1 ≤ n, q ≤ 2 ⋅ 10^5) — the number of elements of the array and the number of queries perfomed on it. The second line contains n integer numbers a_1, a_2, ..., a_n (0 ≤ a_i ≤ q) — the resulting array. If element at some position j is equal to 0 then the value of element at this position can be any integer from 1 to q. Output Print "YES" if the array a can be obtained by performing q queries. Segments (l_i, r_i) (1 ≤ l_i ≤ r_i ≤ n) are chosen separately for each query. Every position from 1 to n should be covered by at least one segment. Otherwise print "NO". If some array can be obtained then print n integers on the second line — the i-th number should be equal to the i-th element of the resulting array and should have value from 1 to q. This array should be obtainable by performing exactly q queries. If there are multiple possible arrays then print any of them. Examples Input 4 3 1 0 2 3 Output YES 1 2 2 3 Input 3 10 10 10 10 Output YES 10 10 10 Input 5 6 6 5 6 2 2 Output NO Input 3 5 0 0 0 Output YES 5 4 2 Note In the first example you can also replace 0 with 1 but not with 3. In the second example it doesn't really matter what segments to choose until query 10 when the segment is (1, 3). The third example showcases the fact that the order of queries can't be changed, you can't firstly set (1, 3) to 6 and after that change (2, 2) to 5. The segment of 5 should be applied before segment of 6. There is a lot of correct resulting arrays for the fourth example. The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. There are n pillars aligned in a row and numbered from 1 to n. Initially each pillar contains exactly one disk. The i-th pillar contains a disk having radius a_i. You can move these disks from one pillar to another. You can take a disk from pillar i and place it on top of pillar j if all these conditions are met: 1. there is no other pillar between pillars i and j. Formally, it means that \|i - j\| = 1; 2. pillar i contains exactly one disk; 3. either pillar j contains no disks, or the topmost disk on pillar j has radius strictly greater than the radius of the disk you move. When you place a disk on a pillar that already has some disks on it, you put the new disk on top of previously placed disks, so the new disk will be used to check the third condition if you try to place another disk on the same pillar. You may take any disk and place it on other pillar any number of times, provided that every time you do it, all three aforementioned conditions are met. Now you wonder, is it possible to place all n disks on the same pillar simultaneously? Input The first line contains one integer n (3 ≤ n ≤ 2 ⋅ 10^5) — the number of pillars. The second line contains n integers a_1, a_2, ..., a_i (1 ≤ a_i ≤ n), where a_i is the radius of the disk initially placed on the i-th pillar. All numbers a_i are distinct. Output Print YES if it is possible to place all the disks on the same pillar simultaneously, and NO otherwise. You may print each letter in any case (YES, yes, Yes will all be recognized as positive answer, NO, no and nO will all be recognized as negative answer). Examples Input 4 1 3 4 2 Output YES Input 3 3 1 2 Output NO Note In the first case it is possible to place all disks on pillar 3 using the following sequence of actions: 1. take the disk with radius 3 from pillar 2 and place it on top of pillar 3; 2. take the disk with radius 1 from pillar 1 and place it on top of pillar 2; 3. take the disk with radius 2 from pillar 4 and place it on top of pillar 3; 4. take the disk with radius 1 from pillar 2 and place it on top of pillar 3. The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Petya's friends made him a birthday present — a bracket sequence. Petya was quite disappointed with his gift, because he dreamed of correct bracket sequence, yet he told his friends nothing about his dreams and decided to fix present himself. To make everything right, Petya is going to move at most one bracket from its original place in the sequence to any other position. Reversing the bracket (e.g. turning "(" into ")" or vice versa) isn't allowed. We remind that bracket sequence s is called correct if: * s is empty; * s is equal to "(t)", where t is correct bracket sequence; * s is equal to t_1 t_2, i.e. concatenation of t_1 and t_2, where t_1 and t_2 are correct bracket sequences. For example, "(()())", "()" are correct, while ")(" and "())" are not. Help Petya to fix his birthday present and understand whether he can move one bracket so that the sequence becomes correct. Input First of line of input contains a single number n (1 ≤ n ≤ 200 000) — length of the sequence which Petya received for his birthday. Second line of the input contains bracket sequence of length n, containing symbols "(" and ")". Output Print "Yes" if Petya can make his sequence correct moving at most one bracket. Otherwise print "No". Examples Input 2 )( Output Yes Input 3 (() Output No Input 2 () Output Yes Input 10 )))))((((( Output No Note In the first example, Petya can move first bracket to the end, thus turning the sequence into "()", which is correct bracket sequence. In the second example, there is no way to move at most one bracket so that the sequence becomes correct. In the third example, the sequence is already correct and there's no need to move brackets. The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Recall that a binary search tree is a rooted binary tree, whose nodes each store a key and each have at most two distinguished subtrees, left and right. The key in each node must be greater than any key stored in the left subtree, and less than any key stored in the right subtree. The depth of a vertex is the number of edges on the simple path from the vertex to the root. In particular, the depth of the root is 0. Let's call a binary search tree perfectly balanced if there doesn't exist a binary search tree with the same number of vertices that has a strictly smaller sum of depths of its vertices. Let's call a binary search tree with integer keys striped if both of the following conditions are satisfied for every vertex v: * If v has a left subtree whose root is u, then the parity of the key of v is different from the parity of the key of u. * If v has a right subtree whose root is w, then the parity of the key of v is the same as the parity of the key of w. You are given a single integer n. Find the number of perfectly balanced striped binary search trees with n vertices that have distinct integer keys between 1 and n, inclusive. Output this number modulo 998 244 353. Input The only line contains a single integer n (1 ≤ n ≤ 10^6), denoting the required number of vertices. Output Output the number of perfectly balanced striped binary search trees with n vertices and distinct integer keys between 1 and n, inclusive, modulo 998 244 353. Examples Input 4 Output 1 Input 3 Output 0 Note In the first example, this is the only tree that satisfies the conditions: <image> In the second example, here are various trees that don't satisfy some condition: <image> The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Santa Claus has received letters from n different kids throughout this year. Of course, each kid wants to get some presents from Santa: in particular, the i-th kid asked Santa to give them one of k_i different items as a present. Some items could have been asked by multiple kids. Santa is really busy, so he wants the New Year Bot to choose the presents for all children. Unfortunately, the Bot's algorithm of choosing presents is bugged. To choose a present for some kid, the Bot does the following: * choose one kid x equiprobably among all n kids; * choose some item y equiprobably among all k_x items kid x wants; * choose a kid z who will receive the present equipropably among all n kids (this choice is independent of choosing x and y); the resulting triple (x, y, z) is called the decision of the Bot. If kid z listed item y as an item they want to receive, then the decision valid. Otherwise, the Bot's choice is invalid. Santa is aware of the bug, but he can't estimate if this bug is really severe. To do so, he wants to know the probability that one decision generated according to the aforementioned algorithm is valid. Can you help him? Input The first line contains one integer n (1 ≤ n ≤ 10^6) — the number of kids who wrote their letters to Santa. Then n lines follow, the i-th of them contains a list of items wanted by the i-th kid in the following format: k_i a_{i, 1} a_{i, 2} ... a_{i, k_i} (1 ≤ k_i, a_{i, j} ≤ 10^6), where k_i is the number of items wanted by the i-th kid, and a_{i, j} are the items themselves. No item is contained in the same list more than once. It is guaranteed that ∑ _{i = 1}^{n} k_i ≤ 10^6. Output Print the probatility that the Bot produces a valid decision as follows: Let this probability be represented as an irreducible fraction x/y. You have to print x ⋅ y^{-1} mod 998244353, where y^{-1} is the inverse element of y modulo 998244353 (such integer that y ⋅ y^{-1} has remainder 1 modulo 998244353). Examples Input 2 2 2 1 1 1 Output 124780545 Input 5 2 1 2 2 3 1 3 2 4 3 2 1 4 3 4 3 2 Output 798595483 The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Anna and Maria are in charge of the math club for junior students. When the club gathers together, the students behave badly. They've brought lots of shoe laces to the club and got tied with each other. Specifically, each string ties together two students. Besides, if two students are tied, then the lace connects the first student with the second one as well as the second student with the first one. To restore order, Anna and Maria do the following. First, for each student Anna finds out what other students he is tied to. If a student is tied to exactly one other student, Anna reprimands him. Then Maria gathers in a single group all the students who have been just reprimanded. She kicks them out from the club. This group of students immediately leaves the club. These students takes with them the laces that used to tie them. Then again for every student Anna finds out how many other students he is tied to and so on. And they do so until Anna can reprimand at least one student. Determine how many groups of students will be kicked out of the club. Input The first line contains two integers n and m — the initial number of students and laces (<image>). The students are numbered from 1 to n, and the laces are numbered from 1 to m. Next m lines each contain two integers a and b — the numbers of students tied by the i-th lace (1 ≤ a, b ≤ n, a ≠ b). It is guaranteed that no two students are tied with more than one lace. No lace ties a student to himself. Output Print the single number — the number of groups of students that will be kicked out from the club. Examples Input 3 3 1 2 2 3 3 1 Output 0 Input 6 3 1 2 2 3 3 4 Output 2 Input 6 5 1 4 2 4 3 4 5 4 6 4 Output 1 Note In the first sample Anna and Maria won't kick out any group of students — in the initial position every student is tied to two other students and Anna won't be able to reprimand anyone. In the second sample four students are tied in a chain and two more are running by themselves. First Anna and Maria kick out the two students from both ends of the chain (1 and 4), then — two other students from the chain (2 and 3). At that the students who are running by themselves will stay in the club. In the third sample Anna and Maria will momentarily kick out all students except for the fourth one and the process stops at that point. The correct answer is one. The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Denis, after buying flowers and sweets (you will learn about this story in the next task), went to a date with Nastya to ask her to become a couple. Now, they are sitting in the cafe and finally... Denis asks her to be together, but ... Nastya doesn't give any answer. The poor boy was very upset because of that. He was so sad that he punched some kind of scoreboard with numbers. The numbers are displayed in the same way as on an electronic clock: each digit position consists of 7 segments, which can be turned on or off to display different numbers. The picture shows how all 10 decimal digits are displayed: <image> After the punch, some segments stopped working, that is, some segments might stop glowing if they glowed earlier. But Denis remembered how many sticks were glowing and how many are glowing now. Denis broke exactly k segments and he knows which sticks are working now. Denis came up with the question: what is the maximum possible number that can appear on the board if you turn on exactly k sticks (which are off now)? It is allowed that the number includes leading zeros. Input The first line contains integer n (1 ≤ n ≤ 2000) — the number of digits on scoreboard and k (0 ≤ k ≤ 2000) — the number of segments that stopped working. The next n lines contain one binary string of length 7, the i-th of which encodes the i-th digit of the scoreboard. Each digit on the scoreboard consists of 7 segments. We number them, as in the picture below, and let the i-th place of the binary string be 0 if the i-th stick is not glowing and 1 if it is glowing. Then a binary string of length 7 will specify which segments are glowing now. <image> Thus, the sequences "1110111", "0010010", "1011101", "1011011", "0111010", "1101011", "1101111", "1010010", "1111111", "1111011" encode in sequence all digits from 0 to 9 inclusive. Output Output a single number consisting of n digits — the maximum number that can be obtained if you turn on exactly k sticks or -1, if it is impossible to turn on exactly k sticks so that a correct number appears on the scoreboard digits. Examples Input 1 7 0000000 Output 8 Input 2 5 0010010 0010010 Output 97 Input 3 5 0100001 1001001 1010011 Output -1 Note In the first test, we are obliged to include all 7 sticks and get one 8 digit on the scoreboard. In the second test, we have sticks turned on so that units are formed. For 5 of additionally included sticks, you can get the numbers 07, 18, 34, 43, 70, 79, 81 and 97, of which we choose the maximum — 97. In the third test, it is impossible to turn on exactly 5 sticks so that a sequence of numbers appears on the scoreboard. The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. A permutation of length n is an array consisting of n distinct integers from 1 to n in arbitrary order. For example, [2,3,1,5,4] is a permutation, but [1,2,2] is not a permutation (2 appears twice in the array) and [1,3,4] is also not a permutation (n=3 but there is 4 in the array). Let p be any permutation of length n. We define the fingerprint F(p) of p as the sorted array of sums of adjacent elements in p. More formally, $$$F(p)=sort([p_1+p_2,p_2+p_3,…,p_{n-1}+p_n]).$$$ For example, if n=4 and p=[1,4,2,3], then the fingerprint is given by F(p)=sort([1+4,4+2,2+3])=sort([5,6,5])=[5,5,6]. You are given a permutation p of length n. Your task is to find a different permutation p' with the same fingerprint. Two permutations p and p' are considered different if there is some index i such that p_i ≠ p'_i. Input Each test contains multiple test cases. The first line contains the number of test cases t (1 ≤ t ≤ 668). Description of the test cases follows. The first line of each test case contains a single integer n (2≤ n≤ 100) — the length of the permutation. The second line of each test case contains n integers p_1,…,p_n (1≤ p_i≤ n). It is guaranteed that p is a permutation. Output For each test case, output n integers p'_1,…, p'_n — a permutation such that p'≠ p and F(p')=F(p). We can prove that for every permutation satisfying the input constraints, a solution exists. If there are multiple solutions, you may output any. Example Input 3 2 1 2 6 2 1 6 5 4 3 5 2 4 3 1 5 Output 2 1 1 2 5 6 3 4 3 1 5 2 4 Note In the first test case, F(p)=sort([1+2])=[3]. And F(p')=sort([2+1])=[3]. In the second test case, F(p)=sort([2+1,1+6,6+5,5+4,4+3])=sort([3,7,11,9,7])=[3,7,7,9,11]. And F(p')=sort([1+2,2+5,5+6,6+3,3+4])=sort([3,7,11,9,7])=[3,7,7,9,11]. In the third test case, F(p)=sort([2+4,4+3,3+1,1+5])=sort([6,7,4,6])=[4,6,6,7]. And F(p')=sort([3+1,1+5,5+2,2+4])=sort([4,6,7,6])=[4,6,6,7]. The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Initially, you have the array a consisting of one element 1 (a = [1]). In one move, you can do one of the following things: * Increase some (single) element of a by 1 (choose some i from 1 to the current length of a and increase a_i by one); * Append the copy of some (single) element of a to the end of the array (choose some i from 1 to the current length of a and append a_i to the end of the array). For example, consider the sequence of five moves: 1. You take the first element a_1, append its copy to the end of the array and get a = [1, 1]. 2. You take the first element a_1, increase it by 1 and get a = [2, 1]. 3. You take the second element a_2, append its copy to the end of the array and get a = [2, 1, 1]. 4. You take the first element a_1, append its copy to the end of the array and get a = [2, 1, 1, 2]. 5. You take the fourth element a_4, increase it by 1 and get a = [2, 1, 1, 3]. Your task is to find the minimum number of moves required to obtain the array with the sum at least n. You have to answer t independent test cases. Input The first line of the input contains one integer t (1 ≤ t ≤ 1000) — the number of test cases. Then t test cases follow. The only line of the test case contains one integer n (1 ≤ n ≤ 10^9) — the lower bound on the sum of the array. Output For each test case, print the answer: the minimum number of moves required to obtain the array with the sum at least n. Example Input 5 1 5 42 1337 1000000000 Output 0 3 11 72 63244 The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. The Super Duper Secret Meeting of the Super Duper Secret Military Squad takes place in a Super Duper Secret Place. The place is an infinite plane with introduced Cartesian coordinate system. The meeting table is represented as a rectangle whose sides are parallel to the coordinate axes and whose vertexes are located at the integer points of the plane. At each integer point which belongs to the table perimeter there is a chair in which a general sits. Some points on the plane contain radiators for the generals not to freeze in winter. Each radiator is characterized by the number ri — the radius of the area this radiator can heat. That is, if the distance between some general and the given radiator is less than or equal to ri, than the general feels comfortable and warm. Here distance is defined as Euclidean distance, so the distance between points (x1, y1) and (x2, y2) is <image> Each general who is located outside the radiators' heating area can get sick. Thus, you should bring him a warm blanket. Your task is to count the number of warm blankets you should bring to the Super Duper Secret Place. The generals who are already comfortable do not need a blanket. Also the generals never overheat, ever if they are located in the heating area of several radiators. The radiators can be located at any integer points on the plane, even inside the rectangle (under the table) or on the perimeter (directly under some general). Even in this case their radius does not change. Input The first input line contains coordinates of two opposite table corners xa, ya, xb, yb (xa ≠ xb, ya ≠ yb). The second line contains integer n — the number of radiators (1 ≤ n ≤ 103). Then n lines contain the heaters' coordinates as "xi yi ri", the numbers are separated by spaces. All input data numbers are integers. The absolute value of all coordinates does not exceed 1000, 1 ≤ ri ≤ 1000. Several radiators can be located at the same point. Output Print the only number — the number of blankets you should bring. Examples Input 2 5 4 2 3 3 1 2 5 3 1 1 3 2 Output 4 Input 5 2 6 3 2 6 2 2 6 5 3 Output 0 Note In the first sample the generals are sitting at points: (2, 2), (2, 3), (2, 4), (2, 5), (3, 2), (3, 5), (4, 2), (4, 3), (4, 4), (4, 5). Among them, 4 generals are located outside the heating range. They are the generals at points: (2, 5), (3, 5), (4, 4), (4, 5). In the second sample the generals are sitting at points: (5, 2), (5, 3), (6, 2), (6, 3). All of them are located inside the heating range. The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Let's define a multiplication operation between a string a and a positive integer x: a ⋅ x is the string that is a result of writing x copies of a one after another. For example, "abc" ⋅~2~= "abcabc", "a" ⋅~5~= "aaaaa". A string a is divisible by another string b if there exists an integer x such that b ⋅ x = a. For example, "abababab" is divisible by "ab", but is not divisible by "ababab" or "aa". LCM of two strings s and t (defined as LCM(s, t)) is the shortest non-empty string that is divisible by both s and t. You are given two strings s and t. Find LCM(s, t) or report that it does not exist. It can be shown that if LCM(s, t) exists, it is unique. Input The first line contains one integer q (1 ≤ q ≤ 2000) — the number of test cases. Each test case consists of two lines, containing strings s and t (1 ≤ \|s\|, \|t\| ≤ 20). Each character in each of these strings is either 'a' or 'b'. Output For each test case, print LCM(s, t) if it exists; otherwise, print -1. It can be shown that if LCM(s, t) exists, it is unique. Example Input 3 baba ba aa aaa aba ab Output baba aaaaaa -1 Note In the first test case, "baba" = "baba" ⋅~1~= "ba" ⋅~2. In the second test case, "aaaaaa" = "aa" ⋅~3~= "aaa" ⋅~2. The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. You are given a string s of length n consisting only of the characters 0 and 1. You perform the following operation until the string becomes empty: choose some consecutive substring of equal characters, erase it from the string and glue the remaining two parts together (any of them can be empty) in the same order. For example, if you erase the substring 111 from the string 111110, you will get the string 110. When you delete a substring of length l, you get a ⋅ l + b points. Your task is to calculate the maximum number of points that you can score in total, if you have to make the given string empty. Input The first line contains a single integer t (1 ≤ t ≤ 2000) — the number of testcases. The first line of each testcase contains three integers n, a and b (1 ≤ n ≤ 100; -100 ≤ a, b ≤ 100) — the length of the string s and the parameters a and b. The second line contains the string s. The string s consists only of the characters 0 and 1. Output For each testcase, print a single integer — the maximum number of points that you can score. Example Input 3 3 2 0 000 5 -2 5 11001 6 1 -4 100111 Output 6 15 -2 Note In the first example, it is enough to delete the entire string, then we will get 2 ⋅ 3 + 0 = 6 points. In the second example, if we delete characters one by one, then for each deleted character we will get (-2) ⋅ 1 + 5 = 3 points, i. e. 15 points in total. In the third example, we can delete the substring 00 from the string 100111, we get 1 ⋅ 2 + (-4) = -2 points, and the string will be equal to 1111, removing it entirely we get 1 ⋅ 4 + (-4) = 0 points. In total, we got -2 points for 2 operations. The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. You are given two polynomials: * P(x) = a0·xn + a1·xn - 1 + ... + an - 1·x + an and * Q(x) = b0·xm + b1·xm - 1 + ... + bm - 1·x + bm. Calculate limit <image>. Input The first line contains two space-separated integers n and m (0 ≤ n, m ≤ 100) — degrees of polynomials P(x) and Q(x) correspondingly. The second line contains n + 1 space-separated integers — the factors of polynomial P(x): a0, a1, ..., an - 1, an ( - 100 ≤ ai ≤ 100, a0 ≠ 0). The third line contains m + 1 space-separated integers — the factors of polynomial Q(x): b0, b1, ..., bm - 1, bm ( - 100 ≤ bi ≤ 100, b0 ≠ 0). Output If the limit equals + ∞, print "Infinity" (without quotes). If the limit equals - ∞, print "-Infinity" (without the quotes). If the value of the limit equals zero, print "0/1" (without the quotes). Otherwise, print an irreducible fraction — the value of limit <image>, in the format "p/q" (without the quotes), where p is the — numerator, q (q > 0) is the denominator of the fraction. Examples Input 2 1 1 1 1 2 5 Output Infinity Input 1 0 -1 3 2 Output -Infinity Input 0 1 1 1 0 Output 0/1 Input 2 2 2 1 6 4 5 -7 Output 1/2 Input 1 1 9 0 -5 2 Output -9/5 Note Let's consider all samples: 1. <image> 2. <image> 3. <image> 4. <image> 5. <image> You can learn more about the definition and properties of limits if you follow the link: http://en.wikipedia.org/wiki/Limit_of_a_function The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. A bracket sequence is called regular if it is possible to obtain correct arithmetic expression by inserting characters «+» and «1» into this sequence. For example, sequences «(())()», «()» and «(()(()))» are regular, while «)(», «(()» and «(()))(» are not. One day Johnny got bracket sequence. He decided to remove some of the brackets from it in order to obtain a regular bracket sequence. What is the maximum length of a regular bracket sequence which can be obtained? Input Input consists of a single line with non-empty string of «(» and «)» characters. Its length does not exceed 106. Output Output the maximum possible length of a regular bracket sequence. Examples Input (()))( Output 4 Input ((()()) Output 6 The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Iahub has drawn a set of n points in the cartesian plane which he calls "special points". A quadrilateral is a simple polygon without self-intersections with four sides (also called edges) and four vertices (also called corners). Please note that a quadrilateral doesn't have to be convex. A special quadrilateral is one which has all four vertices in the set of special points. Given the set of special points, please calculate the maximal area of a special quadrilateral. Input The first line contains integer n (4 ≤ n ≤ 300). Each of the next n lines contains two integers: xi, yi ( - 1000 ≤ xi, yi ≤ 1000) — the cartesian coordinates of ith special point. It is guaranteed that no three points are on the same line. It is guaranteed that no two points coincide. Output Output a single real number — the maximal area of a special quadrilateral. The answer will be considered correct if its absolute or relative error does't exceed 10 - 9. Examples Input 5 0 0 0 4 4 0 4 4 2 3 Output 16.000000 Note In the test example we can choose first 4 points to be the vertices of the quadrilateral. They form a square by side 4, so the area is 4·4 = 16. The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. A group of n schoolboys decided to ride bikes. As nobody of them has a bike, the boys need to rent them. The renting site offered them m bikes. The renting price is different for different bikes, renting the j-th bike costs pj rubles. In total, the boys' shared budget is a rubles. Besides, each of them has his own personal money, the i-th boy has bi personal rubles. The shared budget can be spent on any schoolchildren arbitrarily, but each boy's personal money can be spent on renting only this boy's bike. Each boy can rent at most one bike, one cannot give his bike to somebody else. What maximum number of schoolboys will be able to ride bikes? What minimum sum of personal money will they have to spend in total to let as many schoolchildren ride bikes as possible? Input The first line of the input contains three integers n, m and a (1 ≤ n, m ≤ 105; 0 ≤ a ≤ 109). The second line contains the sequence of integers b1, b2, ..., bn (1 ≤ bi ≤ 104), where bi is the amount of the i-th boy's personal money. The third line contains the sequence of integers p1, p2, ..., pm (1 ≤ pj ≤ 109), where pj is the price for renting the j-th bike. Output Print two integers r and s, where r is the maximum number of schoolboys that can rent a bike and s is the minimum total personal money needed to rent r bikes. If the schoolchildren cannot rent any bikes, then r = s = 0. Examples Input 2 2 10 5 5 7 6 Output 2 3 Input 4 5 2 8 1 1 2 6 3 7 5 2 Output 3 8 Note In the first sample both schoolchildren can rent a bike. For instance, they can split the shared budget in half (5 rubles each). In this case one of them will have to pay 1 ruble from the personal money and the other one will have to pay 2 rubles from the personal money. In total, they spend 3 rubles of their personal money. This way of distribution of money minimizes the amount of spent personal money. The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. George is a cat, so he really likes to play. Most of all he likes to play with his array of positive integers b. During the game, George modifies the array by using special changes. Let's mark George's current array as b1, b2, ..., b\|b\| (record \|b\| denotes the current length of the array). Then one change is a sequence of actions: * Choose two distinct indexes i and j (1 ≤ i, j ≤ \|b\|; i ≠ j), such that bi ≥ bj. * Get number v = concat(bi, bj), where concat(x, y) is a number obtained by adding number y to the end of the decimal record of number x. For example, concat(500, 10) = 50010, concat(2, 2) = 22. * Add number v to the end of the array. The length of the array will increase by one. * Remove from the array numbers with indexes i and j. The length of the array will decrease by two, and elements of the array will become re-numbered from 1 to current length of the array. George played for a long time with his array b and received from array b an array consisting of exactly one number p. Now George wants to know: what is the maximum number of elements array b could contain originally? Help him find this number. Note that originally the array could contain only positive integers. Input The first line of the input contains a single integer p (1 ≤ p < 10100000). It is guaranteed that number p doesn't contain any leading zeroes. Output Print an integer — the maximum number of elements array b could contain originally. Examples Input 9555 Output 4 Input 10000000005 Output 2 Input 800101 Output 3 Input 45 Output 1 Input 1000000000000001223300003342220044555 Output 17 Input 19992000 Output 1 Input 310200 Output 2 Note Let's consider the test examples: * Originally array b can be equal to {5, 9, 5, 5}. The sequence of George's changes could have been: {5, 9, 5, 5} → {5, 5, 95} → {95, 55} → {9555}. * Originally array b could be equal to {1000000000, 5}. Please note that the array b cannot contain zeros. * Originally array b could be equal to {800, 10, 1}. * Originally array b could be equal to {45}. It cannot be equal to {4, 5}, because George can get only array {54} from this array in one operation. Note that the numbers can be very large. The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Pasha has a positive integer a without leading zeroes. Today he decided that the number is too small and he should make it larger. Unfortunately, the only operation Pasha can do is to swap two adjacent decimal digits of the integer. Help Pasha count the maximum number he can get if he has the time to make at most k swaps. Input The single line contains two integers a and k (1 ≤ a ≤ 1018; 0 ≤ k ≤ 100). Output Print the maximum number that Pasha can get if he makes at most k swaps. Examples Input 1990 1 Output 9190 Input 300 0 Output 300 Input 1034 2 Output 3104 Input 9090000078001234 6 Output 9907000008001234 The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. One day Vasya came across three Berland coins. They didn't have any numbers that's why Vasya didn't understand how their denominations differ. He supposed that if one coin is heavier than the other one, then it should be worth more. Vasya weighed all the three pairs of coins on pan balance scales and told you the results. Find out how the deminations of the coins differ or if Vasya has a mistake in the weighting results. No two coins are equal. Input The input data contains the results of all the weighting, one result on each line. It is guaranteed that every coin pair was weighted exactly once. Vasya labelled the coins with letters «A», «B» and «C». Each result is a line that appears as (letter)(> or < sign)(letter). For example, if coin "A" proved lighter than coin "B", the result of the weighting is A<B. Output It the results are contradictory, print Impossible. Otherwise, print without spaces the rearrangement of letters «A», «B» and «C» which represent the coins in the increasing order of their weights. Examples Input A>B C<B A>C Output CBA Input A<B B>C C>A Output ACB The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Let's define the sum of two permutations p and q of numbers 0, 1, ..., (n - 1) as permutation <image>, where Perm(x) is the x-th lexicographically permutation of numbers 0, 1, ..., (n - 1) (counting from zero), and Ord(p) is the number of permutation p in the lexicographical order. For example, Perm(0) = (0, 1, ..., n - 2, n - 1), Perm(n! - 1) = (n - 1, n - 2, ..., 1, 0) Misha has two permutations, p and q. Your task is to find their sum. Permutation a = (a0, a1, ..., an - 1) is called to be lexicographically smaller than permutation b = (b0, b1, ..., bn - 1), if for some k following conditions hold: a0 = b0, a1 = b1, ..., ak - 1 = bk - 1, ak < bk. Input The first line contains an integer n (1 ≤ n ≤ 200 000). The second line contains n distinct integers from 0 to n - 1, separated by a space, forming permutation p. The third line contains n distinct integers from 0 to n - 1, separated by spaces, forming permutation q. Output Print n distinct integers from 0 to n - 1, forming the sum of the given permutations. Separate the numbers by spaces. Examples Input 2 0 1 0 1 Output 0 1 Input 2 0 1 1 0 Output 1 0 Input 3 1 2 0 2 1 0 Output 1 0 2 Note Permutations of numbers from 0 to 1 in the lexicographical order: (0, 1), (1, 0). In the first sample Ord(p) = 0 and Ord(q) = 0, so the answer is <image>. In the second sample Ord(p) = 0 and Ord(q) = 1, so the answer is <image>. Permutations of numbers from 0 to 2 in the lexicographical order: (0, 1, 2), (0, 2, 1), (1, 0, 2), (1, 2, 0), (2, 0, 1), (2, 1, 0). In the third sample Ord(p) = 3 and Ord(q) = 5, so the answer is <image>. The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. There are many anime that are about "love triangles": Alice loves Bob, and Charlie loves Bob as well, but Alice hates Charlie. You are thinking about an anime which has n characters. The characters are labeled from 1 to n. Every pair of two characters can either mutually love each other or mutually hate each other (there is no neutral state). You hate love triangles (A-B are in love and B-C are in love, but A-C hate each other), and you also hate it when nobody is in love. So, considering any three characters, you will be happy if exactly one pair is in love (A and B love each other, and C hates both A and B), or if all three pairs are in love (A loves B, B loves C, C loves A). You are given a list of m known relationships in the anime. You know for sure that certain pairs love each other, and certain pairs hate each other. You're wondering how many ways you can fill in the remaining relationships so you are happy with every triangle. Two ways are considered different if two characters are in love in one way but hate each other in the other. Print this count modulo 1 000 000 007. Input The first line of input will contain two integers n, m (3 ≤ n ≤ 100 000, 0 ≤ m ≤ 100 000). The next m lines will contain the description of the known relationships. The i-th line will contain three integers ai, bi, ci. If ci is 1, then ai and bi are in love, otherwise, they hate each other (1 ≤ ai, bi ≤ n, ai ≠ bi, <image>). Each pair of people will be described no more than once. Output Print a single integer equal to the number of ways to fill in the remaining pairs so that you are happy with every triangle modulo 1 000 000 007. Examples Input 3 0 Output 4 Input 4 4 1 2 1 2 3 1 3 4 0 4 1 0 Output 1 Input 4 4 1 2 1 2 3 1 3 4 0 4 1 1 Output 0 Note In the first sample, the four ways are to: * Make everyone love each other * Make 1 and 2 love each other, and 3 hate 1 and 2 (symmetrically, we get 3 ways from this). In the second sample, the only possible solution is to make 1 and 3 love each other and 2 and 4 hate each other. The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. In Absurdistan, there are n towns (numbered 1 through n) and m bidirectional railways. There is also an absurdly simple road network — for each pair of different towns x and y, there is a bidirectional road between towns x and y if and only if there is no railway between them. Travelling to a different town using one railway or one road always takes exactly one hour. A train and a bus leave town 1 at the same time. They both have the same destination, town n, and don't make any stops on the way (but they can wait in town n). The train can move only along railways and the bus can move only along roads. You've been asked to plan out routes for the vehicles; each route can use any road/railway multiple times. One of the most important aspects to consider is safety — in order to avoid accidents at railway crossings, the train and the bus must not arrive at the same town (except town n) simultaneously. Under these constraints, what is the minimum number of hours needed for both vehicles to reach town n (the maximum of arrival times of the bus and the train)? Note, that bus and train are not required to arrive to the town n at the same moment of time, but are allowed to do so. Input The first line of the input contains two integers n and m (2 ≤ n ≤ 400, 0 ≤ m ≤ n(n - 1) / 2) — the number of towns and the number of railways respectively. Each of the next m lines contains two integers u and v, denoting a railway between towns u and v (1 ≤ u, v ≤ n, u ≠ v). You may assume that there is at most one railway connecting any two towns. Output Output one integer — the smallest possible time of the later vehicle's arrival in town n. If it's impossible for at least one of the vehicles to reach town n, output - 1. Examples Input 4 2 1 3 3 4 Output 2 Input 4 6 1 2 1 3 1 4 2 3 2 4 3 4 Output -1 Input 5 5 4 2 3 5 4 5 5 1 1 2 Output 3 Note In the first sample, the train can take the route <image> and the bus can take the route <image>. Note that they can arrive at town 4 at the same time. In the second sample, Absurdistan is ruled by railwaymen. There are no roads, so there's no way for the bus to reach town 4. The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. A MIPT student named Misha has a birthday today, and he decided to celebrate it in his country house in suburban Moscow. n friends came by, and after a typical party they decided to play blind man's buff. The birthday boy gets blindfolded and the other players scatter around the house. The game is played in several rounds. In each round, Misha catches exactly one of his friends and has to guess who it is. The probability of catching the i-th friend does not change between rounds and is equal to pi percent (as we know, it is directly proportional to the amount of alcohol consumed by the i-th friend) and p1 + p2 + ... + pn = 100 holds. Misha has no information about who he caught. After Misha makes an attempt to guess the caught person, the round ends. Even then, Misha isn't told whether he guessed correctly, and a new round begins. The game ends when Misha guesses every friend at least once, that is, there exists such set of rounds k1, k2, ..., kn, that during round number ki Misha caught the i-th friend and guessed him. Misha wants to minimize the expectation of the number of rounds of the game. Despite the fact that at any point in the game Misha has no information about who he has already guessed, his friends are honest, and if they see that the condition for the end of the game is fulfilled, the game ends immediately. Find the expectation of the number of rounds in the game if Misha plays optimally. Input The first line of the input contains a single integer n (1 ≤ n ≤ 100) — the number of Misha's friends. The second line contains n integers pi (<image>), giving the probability to catch the i-th friend in one particular round in percent. Output Print a single real value — the expectation of the number of rounds provided that Misha plays optimally. Your answer will be considered correct if its absolute or relative error does not exceed 10 - 6. Namely: let's assume that your answer is a, and the answer of the jury is b. The checker program will consider your answer correct, if <image>. Examples Input 2 50 50 Output 5.0000000000 Input 4 50 20 20 10 Output 39.2846263444 Note The optimal strategy in the first sample is to guess friends alternately. The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. In an attempt to escape the Mischievous Mess Makers' antics, Farmer John has abandoned his farm and is traveling to the other side of Bovinia. During the journey, he and his k cows have decided to stay at the luxurious Grand Moo-dapest Hotel. The hotel consists of n rooms located in a row, some of which are occupied. Farmer John wants to book a set of k + 1 currently unoccupied rooms for him and his cows. He wants his cows to stay as safe as possible, so he wishes to minimize the maximum distance from his room to the room of his cow. The distance between rooms i and j is defined as \|j - i\|. Help Farmer John protect his cows by calculating this minimum possible distance. Input The first line of the input contains two integers n and k (1 ≤ k < n ≤ 100 000) — the number of rooms in the hotel and the number of cows travelling with Farmer John. The second line contains a string of length n describing the rooms. The i-th character of the string will be '0' if the i-th room is free, and '1' if the i-th room is occupied. It is guaranteed that at least k + 1 characters of this string are '0', so there exists at least one possible choice of k + 1 rooms for Farmer John and his cows to stay in. Output Print the minimum possible distance between Farmer John's room and his farthest cow. Examples Input 7 2 0100100 Output 2 Input 5 1 01010 Output 2 Input 3 2 000 Output 1 Note In the first sample, Farmer John can book room 3 for himself, and rooms 1 and 4 for his cows. The distance to the farthest cow is 2. Note that it is impossible to make this distance 1, as there is no block of three consecutive unoccupied rooms. In the second sample, Farmer John can book room 1 for himself and room 3 for his single cow. The distance between him and his cow is 2. In the third sample, Farmer John books all three available rooms, taking the middle room for himself so that both cows are next to him. His distance from the farthest cow is 1. The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Every year, hundreds of people come to summer camps, they learn new algorithms and solve hard problems. This is your first year at summer camp, and you are asked to solve the following problem. All integers starting with 1 are written in one line. The prefix of these line is "123456789101112131415...". Your task is to print the n-th digit of this string (digits are numbered starting with 1. Input The only line of the input contains a single integer n (1 ≤ n ≤ 1000) — the position of the digit you need to print. Output Print the n-th digit of the line. Examples Input 3 Output 3 Input 11 Output 0 Note In the first sample the digit at position 3 is '3', as both integers 1 and 2 consist on one digit. In the second sample, the digit at position 11 is '0', it belongs to the integer 10. The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Efim just received his grade for the last test. He studies in a special school and his grade can be equal to any positive decimal fraction. First he got disappointed, as he expected a way more pleasant result. Then, he developed a tricky plan. Each second, he can ask his teacher to round the grade at any place after the decimal point (also, he can ask to round to the nearest integer). There are t seconds left till the end of the break, so Efim has to act fast. Help him find what is the maximum grade he can get in no more than t seconds. Note, that he can choose to not use all t seconds. Moreover, he can even choose to not round the grade at all. In this problem, classic rounding rules are used: while rounding number to the n-th digit one has to take a look at the digit n + 1. If it is less than 5 than the n-th digit remain unchanged while all subsequent digits are replaced with 0. Otherwise, if the n + 1 digit is greater or equal to 5, the digit at the position n is increased by 1 (this might also change some other digits, if this one was equal to 9) and all subsequent digits are replaced with 0. At the end, all trailing zeroes are thrown away. For example, if the number 1.14 is rounded to the first decimal place, the result is 1.1, while if we round 1.5 to the nearest integer, the result is 2. Rounding number 1.299996121 in the fifth decimal place will result in number 1.3. Input The first line of the input contains two integers n and t (1 ≤ n ≤ 200 000, 1 ≤ t ≤ 109) — the length of Efim's grade and the number of seconds till the end of the break respectively. The second line contains the grade itself. It's guaranteed that the grade is a positive number, containing at least one digit after the decimal points, and it's representation doesn't finish with 0. Output Print the maximum grade that Efim can get in t seconds. Do not print trailing zeroes. Examples Input 6 1 10.245 Output 10.25 Input 6 2 10.245 Output 10.3 Input 3 100 9.2 Output 9.2 Note In the first two samples Efim initially has grade 10.245. During the first second Efim can obtain grade 10.25, and then 10.3 during the next second. Note, that the answer 10.30 will be considered incorrect. In the third sample the optimal strategy is to not perform any rounding at all. The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Vasya plays The Elder Trolls IV: Oblivon. Oh, those creators of computer games! What they do not come up with! Absolutely unique monsters have been added to the The Elder Trolls IV: Oblivon. One of these monsters is Unkillable Slug. Why it is "Unkillable"? Firstly, because it can be killed with cutting weapon only, so lovers of two-handed amber hammers should find suitable knife themselves. Secondly, it is necessary to make so many cutting strokes to Unkillable Slug. Extremely many. Too many! Vasya has already promoted his character to 80-th level and in order to gain level 81 he was asked to kill Unkillable Slug. The monster has a very interesting shape. It looks like a rectangular parallelepiped with size x × y × z, consisting of undestructable cells 1 × 1 × 1. At one stroke Vasya can cut the Slug along an imaginary grid, i.e. cut with a plane parallel to one of the parallelepiped side. Monster dies when amount of parts it is divided reaches some critical value. All parts of monster do not fall after each cut, they remains exactly on its places. I. e. Vasya can cut several parts with one cut. Vasya wants to know what the maximum number of pieces he can cut the Unkillable Slug into striking him at most k times. Vasya's character uses absolutely thin sword with infinite length. Input The first line of input contains four integer numbers x, y, z, k (1 ≤ x, y, z ≤ 106, 0 ≤ k ≤ 109). Output Output the only number — the answer for the problem. Please, do not use %lld specificator to read or write 64-bit integers in C++. It is preffered to use cout (also you may use %I64d). Examples Input 2 2 2 3 Output 8 Input 2 2 2 1 Output 2 Note In the first sample Vasya make 3 pairwise perpendicular cuts. He cuts monster on two parts with the first cut, then he divides each part on two with the second cut, and finally he divides each of the 4 parts on two. The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Rick and Morty are playing their own version of Berzerk (which has nothing in common with the famous Berzerk game). This game needs a huge space, so they play it with a computer. In this game there are n objects numbered from 1 to n arranged in a circle (in clockwise order). Object number 1 is a black hole and the others are planets. There's a monster in one of the planet. Rick and Morty don't know on which one yet, only that he's not initially in the black hole, but Unity will inform them before the game starts. But for now, they want to be prepared for every possible scenario. <image> Each one of them has a set of numbers between 1 and n - 1 (inclusive). Rick's set is s1 with k1 elements and Morty's is s2 with k2 elements. One of them goes first and the player changes alternatively. In each player's turn, he should choose an arbitrary number like x from his set and the monster will move to his x-th next object from its current position (clockwise). If after his move the monster gets to the black hole he wins. Your task is that for each of monster's initial positions and who plays first determine if the starter wins, loses, or the game will stuck in an infinite loop. In case when player can lose or make game infinity, it more profitable to choose infinity game. Input The first line of input contains a single integer n (2 ≤ n ≤ 7000) — number of objects in game. The second line contains integer k1 followed by k1 distinct integers s1, 1, s1, 2, ..., s1, k1 — Rick's set. The third line contains integer k2 followed by k2 distinct integers s2, 1, s2, 2, ..., s2, k2 — Morty's set 1 ≤ ki ≤ n - 1 and 1 ≤ si, 1, si, 2, ..., si, ki ≤ n - 1 for 1 ≤ i ≤ 2. Output In the first line print n - 1 words separated by spaces where i-th word is "Win" (without quotations) if in the scenario that Rick plays first and monster is initially in object number i + 1 he wins, "Lose" if he loses and "Loop" if the game will never end. Similarly, in the second line print n - 1 words separated by spaces where i-th word is "Win" (without quotations) if in the scenario that Morty plays first and monster is initially in object number i + 1 he wins, "Lose" if he loses and "Loop" if the game will never end. Examples Input 5 2 3 2 3 1 2 3 Output Lose Win Win Loop Loop Win Win Win Input 8 4 6 2 3 4 2 3 6 Output Win Win Win Win Win Win Win Lose Win Lose Lose Win Lose Lose The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. It's one more school day now. Sasha doesn't like classes and is always bored at them. So, each day he invents some game and plays in it alone or with friends. Today he invented one simple game to play with Lena, with whom he shares a desk. The rules are simple. Sasha draws n sticks in a row. After that the players take turns crossing out exactly k sticks from left or right in each turn. Sasha moves first, because he is the inventor of the game. If there are less than k sticks on the paper before some turn, the game ends. Sasha wins if he makes strictly more moves than Lena. Sasha wants to know the result of the game before playing, you are to help him. Input The first line contains two integers n and k (1 ≤ n, k ≤ 1018, k ≤ n) — the number of sticks drawn by Sasha and the number k — the number of sticks to be crossed out on each turn. Output If Sasha wins, print "YES" (without quotes), otherwise print "NO" (without quotes). You can print each letter in arbitrary case (upper of lower). Examples Input 1 1 Output YES Input 10 4 Output NO Note In the first example Sasha crosses out 1 stick, and then there are no sticks. So Lena can't make a move, and Sasha wins. In the second example Sasha crosses out 4 sticks, then Lena crosses out 4 sticks, and after that there are only 2 sticks left. Sasha can't make a move. The players make equal number of moves, so Sasha doesn't win. The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Slava plays his favorite game "Peace Lightning". Now he is flying a bomber on a very specific map. Formally, map is a checkered field of size 1 × n, the cells of which are numbered from 1 to n, in each cell there can be one or several tanks. Slava doesn't know the number of tanks and their positions, because he flies very high, but he can drop a bomb in any cell. All tanks in this cell will be damaged. If a tank takes damage for the first time, it instantly moves to one of the neighboring cells (a tank in the cell n can only move to the cell n - 1, a tank in the cell 1 can only move to the cell 2). If a tank takes damage for the second time, it's counted as destroyed and never moves again. The tanks move only when they are damaged for the first time, they do not move by themselves. Help Slava to destroy all tanks using as few bombs as possible. Input The first line contains a single integer n (2 ≤ n ≤ 100 000) — the size of the map. Output In the first line print m — the minimum number of bombs Slava needs to destroy all tanks. In the second line print m integers k1, k2, ..., km. The number ki means that the i-th bomb should be dropped at the cell ki. If there are multiple answers, you can print any of them. Examples Input 2 Output 3 2 1 2 Input 3 Output 4 2 1 3 2 The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Count the number of distinct sequences a1, a2, ..., an (1 ≤ ai) consisting of positive integers such that gcd(a1, a2, ..., an) = x and <image>. As this number could be large, print the answer modulo 109 + 7. gcd here means the [greatest common divisor](https://en.wikipedia.org/wiki/Greatest_common_divisor). Input The only line contains two positive integers x and y (1 ≤ x, y ≤ 109). Output Print the number of such sequences modulo 109 + 7. Examples Input 3 9 Output 3 Input 5 8 Output 0 Note There are three suitable sequences in the first test: (3, 3, 3), (3, 6), (6, 3). There are no suitable sequences in the second test. The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Dima is a beginner programmer. During his working process, he regularly has to repeat the following operation again and again: to remove every second element from the array. One day he has been bored with easy solutions of this problem, and he has come up with the following extravagant algorithm. Let's consider that initially array contains n numbers from 1 to n and the number i is located in the cell with the index 2i - 1 (Indices are numbered starting from one) and other cells of the array are empty. Each step Dima selects a non-empty array cell with the maximum index and moves the number written in it to the nearest empty cell to the left of the selected one. The process continues until all n numbers will appear in the first n cells of the array. For example if n = 4, the array is changing as follows: <image> You have to write a program that allows you to determine what number will be in the cell with index x (1 ≤ x ≤ n) after Dima's algorithm finishes. Input The first line contains two integers n and q (1 ≤ n ≤ 1018, 1 ≤ q ≤ 200 000), the number of elements in the array and the number of queries for which it is needed to find the answer. Next q lines contain integers xi (1 ≤ xi ≤ n), the indices of cells for which it is necessary to output their content after Dima's algorithm finishes. Output For each of q queries output one integer number, the value that will appear in the corresponding array cell after Dima's algorithm finishes. Examples Input 4 3 2 3 4 Output 3 2 4 Input 13 4 10 5 4 8 Output 13 3 8 9 Note The first example is shown in the picture. In the second example the final array is [1, 12, 2, 8, 3, 11, 4, 9, 5, 13, 6, 10, 7]. The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Petya studies at university. The current academic year finishes with n special days. Petya needs to pass m exams in those special days. The special days in this problem are numbered from 1 to n. There are three values about each exam: * s_i — the day, when questions for the i-th exam will be published, * d_i — the day of the i-th exam (s_i < d_i), * c_i — number of days Petya needs to prepare for the i-th exam. For the i-th exam Petya should prepare in days between s_i and d_i-1, inclusive. There are three types of activities for Petya in each day: to spend a day doing nothing (taking a rest), to spend a day passing exactly one exam or to spend a day preparing for exactly one exam. So he can't pass/prepare for multiple exams in a day. He can't mix his activities in a day. If he is preparing for the i-th exam in day j, then s_i ≤ j < d_i. It is allowed to have breaks in a preparation to an exam and to alternate preparations for different exams in consecutive days. So preparation for an exam is not required to be done in consecutive days. Find the schedule for Petya to prepare for all exams and pass them, or report that it is impossible. Input The first line contains two integers n and m (2 ≤ n ≤ 100, 1 ≤ m ≤ n) — the number of days and the number of exams. Each of the following m lines contains three integers s_i, d_i, c_i (1 ≤ s_i < d_i ≤ n, 1 ≤ c_i ≤ n) — the day, when questions for the i-th exam will be given, the day of the i-th exam, number of days Petya needs to prepare for the i-th exam. Guaranteed, that all the exams will be in different days. Questions for different exams can be given in the same day. It is possible that, in the day of some exam, the questions for other exams are given. Output If Petya can not prepare and pass all the exams, print -1. In case of positive answer, print n integers, where the j-th number is: * (m + 1), if the j-th day is a day of some exam (recall that in each day no more than one exam is conducted), * zero, if in the j-th day Petya will have a rest, * i (1 ≤ i ≤ m), if Petya will prepare for the i-th exam in the day j (the total number of days Petya prepares for each exam should be strictly equal to the number of days needed to prepare for it). Assume that the exams are numbered in order of appearing in the input, starting from 1. If there are multiple schedules, print any of them. Examples Input 5 2 1 3 1 1 5 1 Output 1 2 3 0 3 Input 3 2 1 3 1 1 2 1 Output -1 Input 10 3 4 7 2 1 10 3 8 9 1 Output 2 2 2 1 1 0 4 3 4 4 Note In the first example Petya can, for example, prepare for exam 1 in the first day, prepare for exam 2 in the second day, pass exam 1 in the third day, relax in the fourth day, and pass exam 2 in the fifth day. So, he can prepare and pass all exams. In the second example, there are three days and two exams. So, Petya can prepare in only one day (because in two other days he should pass exams). Then Petya can not prepare and pass all exams. The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Mishka started participating in a programming contest. There are n problems in the contest. Mishka's problem-solving skill is equal to k. Mishka arranges all problems from the contest into a list. Because of his weird principles, Mishka only solves problems from one of the ends of the list. Every time, he chooses which end (left or right) he will solve the next problem from. Thus, each problem Mishka solves is either the leftmost or the rightmost problem in the list. Mishka cannot solve a problem with difficulty greater than k. When Mishka solves the problem, it disappears from the list, so the length of the list decreases by 1. Mishka stops when he is unable to solve any problem from any end of the list. How many problems can Mishka solve? Input The first line of input contains two integers n and k (1 ≤ n, k ≤ 100) — the number of problems in the contest and Mishka's problem-solving skill. The second line of input contains n integers a_1, a_2, ..., a_n (1 ≤ a_i ≤ 100), where a_i is the difficulty of the i-th problem. The problems are given in order from the leftmost to the rightmost in the list. Output Print one integer — the maximum number of problems Mishka can solve. Examples Input 8 4 4 2 3 1 5 1 6 4 Output 5 Input 5 2 3 1 2 1 3 Output 0 Input 5 100 12 34 55 43 21 Output 5 Note In the first example, Mishka can solve problems in the following order: [4, 2, 3, 1, 5, 1, 6, 4] → [2, 3, 1, 5, 1, 6, 4] → [2, 3, 1, 5, 1, 6] → [3, 1, 5, 1, 6] → [1, 5, 1, 6] → [5, 1, 6], so the number of solved problems will be equal to 5. In the second example, Mishka can't solve any problem because the difficulties of problems from both ends are greater than k. In the third example, Mishka's solving skill is so amazing that he can solve all the problems. The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Naruto is a fan of numbers, he likes to play with them. This time he decided to multiply numbers and produce output. After every k elements he decides to find x^m, where x is the multiplication till k elements and m is a random number that naruto takes. Help naruto find the output. As answer could be large, help him, calculate value modulo w. Input T=number of test case N=number of elements he will enter K=every kth element M=Random number W=Modulo A[i]=n elements to be entered. Output Output answer of each test case in one line. Constraint: 0<t<100 0<a[i]<10000000 1<n<101 k<100000 m<100000 w<100000 SAMPLE INPUT 2 5 2 2 5 1 2 3 4 5 5 2 2 5 1 2 3 4 6 SAMPLE OUTPUT 0 4 Explanation In first case n=5,k=2,m=2,w=5; Step 1 1 Step 2: 12=2 since k=2 therefore 2^m=2^2=4; Step 3: 43=12 Step 4: 124=48 since k=2,therefore 48^m=48^2 step 5: 23045=11520 step 6: 11520%5=0 2nd Test case Step 1 1 Step 2: 12=2 since k=2 therefore 2^m=2^2=4; Step 3: 43=12 Step 4: 124=48 since k=2,therefore 48^m=48^2 step 5: 23046=13824 step 6: 13824%5=4 The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. After Joeffrey chocked to death, Cercei blamed Tyrian for his death. After seeing his love, Shae betrayed him, he demanded trial by combat. Now this combat is not with swords but with logic and problem solving. Tywin will ask Tyrian questions and he have to answer them correctly. Tywin asks him this question: There are n lannisters, out of which some are sitting and some are standing. After one hour, some of them will sit according to this rule: Only those lannisters will sit who had both their neighbors sitting the previous hour. For more clarity refer to sample test case. Now, Tywin asks Tyrian to tell who all are standing and who all are sitting after m hours. Note: For lannisters on corners there is only 1 neighbor. [Input] First line contains an integer t denoting no.of test cases. Second line contains two integers n,m denoting number of lannisters and no.of hours respectively. Third line contains n space separated integers either 0 or 1 depending on whether lannister is standing or sitting respectively [Output] For each test case output n space separated integers according to the ques. For more clarity see sample test case. [Constraints] 1 ≤ t ≤ 1000 2 ≤ n ≤ 100 0 ≤ m ≤ 10000 SAMPLE INPUT 2 3 4 1 1 1 6 2 1 0 1 1 0 0 SAMPLE OUTPUT 1 1 1 1 0 0 0 0 0 Explanation 2nd test case: In the first hour the sitting and standing order will be : 0 1 0 0 0 0 because 1st and 6th lannister have no neighbour sitting in the previous hour so they will stand. 2nd lannister have both his neighbours sitting in the previous hour so he will sit. 3rd, 4th and 5th lannister have only one of their neighbour sitting, so they will stand. In the second hour the sitting and standing order will be: 1 0 0 0 0 0 because only 1st lannister have both his neighbour's sitting in the previous hour. Note: For 1st and last person assume only one neighbour. The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Shil is your new boss and he likes palindromes very much. Palindrome is a string that can be read the same way in either direction, from the left to the right and from the right to the left. (ex. madam , aabaa, racecar) Given a string S , beautiful Palindrome is a lexicographical minimum palindrome that can be formed by rearranging all the characters of string S. In order to please your boss, find a beautiful Palindrome that can be formed with help of string S. String x is lexicographically less than string y, if either x is a prefix of y (and x ≠ y), or there exists such i (1 ≤ i ≤ min(\|x\|, \|y\|)), that xi < yi, and for any j (1 ≤ j < i) xj = yj. Here \|a\| denotes the length of the string a. The lexicographic comparison of strings is implemented by operator < in modern programming languages. Input: Only line of input contains string S. All the letters of this string will be in lower letters('a' - 'z'). Output: Output lexicographical minimum Palindrome that can be formed by rearranging all the letters of string S. If no such Palindrome exist for given input, print -1. Constraints: 1≤\|S\|≤100000 SAMPLE INPUT aabcc SAMPLE OUTPUT acbca Explanation After rearranging all the characters , all the palindromes that can be formed are cabac and acbca. Out of these two lexicographical minimum one is acbca. The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. In this problem, we use the 24-hour clock. Takahashi gets up exactly at the time H_1 : M_1 and goes to bed exactly at the time H_2 : M_2. (See Sample Inputs below for clarity.) He has decided to study for K consecutive minutes while he is up. What is the length of the period in which he can start studying? Constraints * 0 \le H_1, H_2 \le 23 * 0 \le M_1, M_2 \le 59 * The time H_1 : M_1 comes before the time H_2 : M_2. * K \ge 1 * Takahashi is up for at least K minutes. * All values in input are integers (without leading zeros). Input Input is given from Standard Input in the following format: H_1 M_1 H_2 M_2 K Output Print the length of the period in which he can start studying, as an integer. Examples Input 10 0 15 0 30 Output 270 Input 10 0 12 0 120 Output 0 The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Given is a positive integer N. Find the number of pairs (A, B) of positive integers not greater than N that satisfy the following condition: * When A and B are written in base ten without leading zeros, the last digit of A is equal to the first digit of B, and the first digit of A is equal to the last digit of B. Constraints * 1 \leq N \leq 2 \times 10^5 * All values in input are integers. Input Input is given from Standard Input in the following format: N Output Print the answer. Examples Input 25 Output 17 Input 1 Output 1 Input 100 Output 108 Input 2020 Output 40812 Input 200000 Output 400000008 The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. We have a sequence of N integers A~=~A_0,~A_1,~...,~A_{N - 1}. Let B be a sequence of K \times N integers obtained by concatenating K copies of A. For example, if A~=~1,~3,~2 and K~=~2, B~=~1,~3,~2,~1,~3,~2. Find the inversion number of B, modulo 10^9 + 7. Here the inversion number of B is defined as the number of ordered pairs of integers (i,~j)~(0 \leq i < j \leq K \times N - 1) such that B_i > B_j. Constraints * All values in input are integers. * 1 \leq N \leq 2000 * 1 \leq K \leq 10^9 * 1 \leq A_i \leq 2000 Input Input is given from Standard Input in the following format: N K A_0 A_1 ... A_{N - 1} Output Print the inversion number of B, modulo 10^9 + 7. Examples Input 2 2 2 1 Output 3 Input 3 5 1 1 1 Output 0 Input 10 998244353 10 9 8 7 5 6 3 4 2 1 Output 185297239 The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. In some other world, today is the day before Christmas Eve. Mr. Takaha is buying N items at a department store. The regular price of the i-th item (1 \leq i \leq N) is p_i yen (the currency of Japan). He has a discount coupon, and can buy one item with the highest price for half the regular price. The remaining N-1 items cost their regular prices. What is the total amount he will pay? Constraints * 2 \leq N \leq 10 * 100 \leq p_i \leq 10000 * p_i is an even number. Input Input is given from Standard Input in the following format: N p_1 p_2 : p_N Output Print the total amount Mr. Takaha will pay. Examples Input 3 4980 7980 6980 Output 15950 Input 4 4320 4320 4320 4320 Output 15120 The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. We have a permutation of the integers from 1 through N, p_1, p_2, .., p_N. We also have M pairs of two integers between 1 and N (inclusive), represented as (x_1,y_1), (x_2,y_2), .., (x_M,y_M). AtCoDeer the deer is going to perform the following operation on p as many times as desired so that the number of i (1 ≤ i ≤ N) such that p_i = i is maximized: * Choose j such that 1 ≤ j ≤ M, and swap p_{x_j} and p_{y_j}. Find the maximum possible number of i such that p_i = i after operations. Constraints * 2 ≤ N ≤ 10^5 * 1 ≤ M ≤ 10^5 * p is a permutation of integers from 1 through N. * 1 ≤ x_j,y_j ≤ N * x_j ≠ y_j * If i ≠ j, \\{x_i,y_i\\} ≠ \\{x_j,y_j\\}. * All values in input are integers. Input Input is given from Standard Input in the following format: N M p_1 p_2 .. p_N x_1 y_1 x_2 y_2 : x_M y_M Output Print the maximum possible number of i such that p_i = i after operations. Examples Input 5 2 5 3 1 4 2 1 3 5 4 Output 2 Input 3 2 3 2 1 1 2 2 3 Output 3 Input 10 8 5 3 6 8 7 10 9 1 2 4 3 1 4 1 5 9 2 5 6 5 3 5 8 9 7 9 Output 8 Input 5 1 1 2 3 4 5 1 5 Output 5 The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Snuke is buying a lamp. The light of the lamp can be adjusted to m levels of brightness, represented by integers from 1 through m, by the two buttons on the remote control. The first button is a "forward" button. When this button is pressed, the brightness level is increased by 1, except when the brightness level is m, in which case the brightness level becomes 1. The second button is a "favorite" button. When this button is pressed, the brightness level becomes the favorite brightness level x, which is set when the lamp is purchased. Snuke is thinking of setting the favorite brightness level x so that he can efficiently adjust the brightness. He is planning to change the brightness n-1 times. In the i-th change, the brightness level is changed from a_i to a_{i+1}. The initial brightness level is a_1. Find the number of times Snuke needs to press the buttons when x is set to minimize this number. Constraints * 2 \leq n,m \leq 10^5 * 1 \leq a_i\leq m * a_i \neq a_{i+1} * n, m and a_i are integers. Input Input is given from Standard Input in the following format: n m a_1 a_2 … a_n Output Print the minimum number of times Snuke needs to press the buttons. Examples Input 4 6 1 5 1 4 Output 5 Input 10 10 10 9 8 7 6 5 4 3 2 1 Output 45 The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. <image> There used to be a game called Joseph's potatoes. Let's say n people are participating. Participants form a circle facing the center and are numbered starting from 1. One hot potato is given to participant n (the large number 30 inside the figure on the left). Participants who are given the potatoes will give the potatoes to the participant on the right. The person passed the mth time is passed to the person on the right and exits the circle (the figure on the left shows the case of m = 9). Each time you hand it over, you will pass one by one, and the last remaining person will be the winner and you will receive potatoes. After n and m are decided, it would be nice to know where you can win before you actually start handing the potatoes. The figure above shows the case of playing this game with the rule that 30 participants exit every 9 people. The large numbers on the inside are the numbers assigned to the participants, and the small numbers on the outside are the numbers that are removed. According to it, it will break out of the circle in the order of 9,18,27,6,16,26, and 21 will remain at the end. That is, 21 is the winner (the smaller number is 30). Enter the number of game participants n and the interval m between the participants who break out of the circle, and create a program that outputs the winner's number. However, m, n <1000. input Given multiple datasets. Each dataset is given in the following format: n m The number of game participants n (integer) and the interval m (integer) between the participants who break out of the circle are given on one line separated by blanks. The input ends with two 0s. The number of datasets does not exceed 50. output For each dataset, output the number (integer) of the winner and the person who will receive the potatoes on one line. Example Input 41 3 30 9 0 0 Output 31 21 The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. At Aizu Riverside Hospital, inpatients walk twice a day for rehabilitation and health promotion. As the number of people trying to recover their physical strength by walking is increasing day by day in order to leave the hospital energetically, the director plans to give a present to the person who walked the longest distance in a day! I launched it. Number of patients n (1 ≤ n ≤ 10000), each patient's number pi (1 ≤ pi ≤ 10000), first walk distance d1i, second walk distance d2i (0 ≤ d1i, d2i ≤ 5000) Create a program that outputs the number of the patient with the longest total walking distance and the distance. However, it is assumed that no patient walks the same distance in a day. Input A sequence of multiple datasets is given as input. The end of the input is indicated by a single line of zeros. Each dataset is given in the following format: n p1 d11 d21 p2 d12 d22 :: pn d1n d2n All inputs are given as integers. The number of datasets does not exceed 50. Output For each input dataset, it prints the number of the patient who walked the longest total distance and the distance walked on one line. Example Input 5 263 2345 2504 1 3210 1985 5000 1501 4132 10000 503 3107 51 1758 2690 3 345 5000 2396 7 3910 1590 6789 2525 3616 0 Output 5000 5633 345 7396 The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Please find the greatest common divisor of two natural numbers. A clue is: The Euclid's algorithm is a way to resolve this task. Input The input file consists of several lines with pairs of two natural numbers in each line. The numbers do not exceed 100000. The number of pairs (datasets) is less than 50. Output Your program has to print the greatest common divisor for each pair of input numbers. Print each result on a new line. Example Input 57 38 60 84 Output 19 12 The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. At 17:00, special agent Jack starts to escape from the enemy camp. There is a cliff in between the camp and the nearest safety zone. Jack has to climb the almost vertical cliff by stepping his feet on the blocks that cover the cliff. The cliff has slippery blocks where Jack has to spend time to take each step. He also has to bypass some blocks that are too loose to support his weight. Your mission is to write a program that calculates the minimum time to complete climbing. Figure D-1 shows an example of cliff data that you will receive. The cliff is covered with square blocks. Jack starts cliff climbing from the ground under the cliff, by stepping his left or right foot on one of the blocks marked with 'S' at the bottom row. The numbers on the blocks are the "slippery levels". It takes t time units for him to safely put his foot on a block marked with t, where 1 ≤ t ≤ 9. He cannot put his feet on blocks marked with 'X'. He completes the climbing when he puts either of his feet on one of the blocks marked with 'T' at the top row. <image> Figure D-1: Example of Cliff Data Jack's movement must meet the following constraints. After putting his left (or right) foot on a block, he can only move his right (or left, respectively) foot. His left foot position (lx, ly) and his right foot position (rx, ry) should satisfy lx < rx and \| lx - rx \| + \| ly - ry \| ≤ 3 . This implies that, given a position of his left foot in Figure D-2 (a), he has to place his right foot on one of the nine blocks marked with blue color. Similarly, given a position of his right foot in Figure D-2 (b), he has to place his left foot on one of the nine blocks marked with blue color. <image> Figure D-2: Possible Placements of Feet Input The input is a sequence of datasets. The end of the input is indicated by a line containing two zeros separated by a space. Each dataset is formatted as follows: > w h > s(1,1) ... s(1,w) > s(2,1) ... s(2,w) > ... > s(h,1) ... s(h,w) > The integers w and h are the width and the height of the matrix data of the cliff. You may assume 2 ≤ w ≤ 30 and 5 ≤ h ≤ 60. Each of the following h lines consists of w characters delimited by a space. The character s(y, x) represents the state of the block at position (x, y) as follows: * 'S': Jack can start cliff climbing from this block. * 'T': Jack finishes climbing when he reaches this block. * 'X': Jack cannot put his feet on this block. * '1' - '9' (= t): Jack has to spend t time units to put either of his feet on this block. You can assume that it takes no time to put a foot on a block marked with 'S' or 'T'. Output For each dataset, print a line only having a decimal integer indicating the minimum time required for the cliff climbing, when Jack can complete it. Otherwise, print a line only having "-1" for the dataset. Each line should not have any characters other than these numbers. Example Input 6 6 4 4 X X T T 4 7 8 2 X 7 3 X X X 1 8 1 2 X X X 6 1 1 2 4 4 7 S S 2 3 X X 2 10 T 1 1 X 1 X 1 X 1 1 1 X 1 X 1 1 1 X S S 2 10 T X 1 X 1 X 1 X 1 1 1 X 1 X 1 1 1 X S S 10 10 T T T T T T T T T T X 2 X X X X X 3 4 X 9 8 9 X X X 2 9 X 9 7 7 X 7 3 X X 8 9 X 8 9 9 9 6 3 X 5 X 5 8 9 9 9 6 X X 5 X 5 8 6 5 4 6 8 X 5 X 5 8 9 3 9 6 8 X 5 X 5 8 3 9 9 6 X X X 5 X S S S S S S S S S S 10 7 2 3 2 3 2 3 2 3 T T 1 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 4 3 2 3 2 3 2 3 2 3 5 3 2 3 1 3 2 3 2 3 5 2 2 3 2 4 2 3 2 3 5 S S 2 3 2 1 2 3 2 3 0 0 Output 12 5 -1 22 12 The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Mr. Yamada Springfield Tanaka was appointed as Deputy Deputy Director of the National Land Readjustment Business Bureau. Currently, his country is in the midst of a major land readjustment, and if this land readjustment can be completed smoothly, his promotion is certain. However, there are many people who are not happy with his career advancement. One such person was Mr. Sato Seabreeze Suzuki. He has planned to pull Mr. Yamada's foot every time. Again, Mr. Sato put pressure on the organization responsible for the actual land readjustment in order to pull back, making the land readjustment results very confusing. Therefore, the result passed to Mr. Yamada was only information about which straight line divided a square land. At the very least, Mr. Yamada is sure to be fired rather than promoted if he doesn't know just how many divisions the square land has. Your job is how many square regions with vertices (-100, -100), (100, -100), (100, 100), (-100, 100) are divided by a given n straight lines. To save Mr. Yamada from the crisis of dismissal by writing a program to find out. Input The input consists of multiple test cases. The first line of each test case is given the integer n representing the number of straight lines (1 <= n <= 100). Subsequent n lines contain four integers x1, y1, x2, and y2, respectively. These integers represent two different points (x1, y1) and (x2, y2) on a straight line. The two points given are always guaranteed to be points on the sides of the square. The n straight lines given are different from each other, and the straight lines do not overlap. Also, the straight line does not overlap the sides of the square. The end of input is represented by n = 0. Output For each test case, print the number of regions divided by n straight lines in one line. Two points with a distance of less than 10-10 can be considered to match. Also, there is no set of intersections P, Q, R such that \| PQ \| <10-10, \| QR \| <10-10 and \| PR \|> = 10-10. Example Input 2 -100 -20 100 20 -20 -100 20 100 2 -100 -20 -20 -100 20 100 100 20 0 Output 4 3 The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. For given two circles $c1$ and $c2$, print the coordinates of the cross points of them. Constraints * The given circle have at least one cross point and have different center coordinates. * $-10,000 \leq c1x, c1y, c2x, c2y \leq 10,000$ * $1 \leq c1r, c2r \leq 10,000$ Input The input is given in the following format. $c1x\; c1y\; c1r$ $c2x\; c2y\; c2r$ $c1x$, $c1y$ and $c1r$ represent the coordinate and radius of the first circle. $c2x$, $c2y$ and $c2r$ represent the coordinate and radius of the second circle. All input values are given in integers. Output Print the coordinates ($x1$, $y1$) and ($x2$, $y2$) of the cross points $p1$ and $p2$ respectively in the following rules. * If there is one cross point, print two coordinates with the same values. * Print the coordinate with smaller $x$ first. In case of a tie, print the coordinate with smaller $y$ first. The output values should be in a decimal fraction with an error less than 0.000001. Examples Input 0 0 2 2 0 2 Output 1.00000000 -1.73205080 1.00000000 1.73205080 Input 0 0 2 0 3 1 Output 0.00000000 2.00000000 0.00000000 2.00000000 The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. For a set $S$ of integers, perform a sequence of the following operations. Note that each value in $S$ must be unique. * insert($x$): Insert $x$ to $S$ and report the number of elements in $S$ after the operation. * find($x$): Report the number of $x$ in $S$ (0 or 1). * delete($x$): Delete $x$ from $S$. * dump($L$, $R$): Print elements $x$ in $S$ such that $L \leq x \leq R$. Constraints * $1 \leq q \leq 200,000$ * $0 \leq x \leq 1,000,000,000$ * The total number of elements printed by dump operations does not exceed $1,000,000$ Input The input is given in the following format. $q$ $query_1$ $query_2$ : $query_q$ Each query $query_i$ is given by 0 $x$ or 1 $x$ or 2 $x$ or 3 $L$ $R$ where the first digits 0, 1, 2 and 3 represent insert, find, delete and dump operations respectively. Output For each insert operation, print the number of elements in $S$. For each find operation, print the number of specified elements in $S$. For each dump operation, print the corresponding elements in ascending order. Print an element in a line. Example Input 9 0 1 0 2 0 3 2 2 1 1 1 2 1 3 0 4 3 2 4 Output 1 2 3 1 0 1 3 3 4 The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Rudolf is on his way to the castle. Before getting into the castle, the security staff asked him a question: Given two binary numbers a and b of length n. How many different ways of swapping two digits in a (only in a, not b) so that bitwise OR of these two numbers will be changed? In other words, let c be the bitwise OR of a and b, you need to find the number of ways of swapping two bits in a so that bitwise OR will not be equal to c. Note that binary numbers can contain leading zeros so that length of each number is exactly n. [Bitwise OR](https://en.wikipedia.org/wiki/Bitwise_operation#OR) is a binary operation. A result is a binary number which contains a one in each digit if there is a one in at least one of the two numbers. For example, 01010_2 OR 10011_2 = 11011_2. Well, to your surprise, you are not Rudolf, and you don't need to help him… You are the security staff! Please find the number of ways of swapping two bits in a so that bitwise OR will be changed. Input The first line contains one integer n (2≤ n≤ 10^5) — the number of bits in each number. The second line contains a binary number a of length n. The third line contains a binary number b of length n. Output Print the number of ways to swap two bits in a so that bitwise OR will be changed. Examples Input 5 01011 11001 Output 4 Input 6 011000 010011 Output 6 Note In the first sample, you can swap bits that have indexes (1, 4), (2, 3), (3, 4), and (3, 5). In the second example, you can swap bits that have indexes (1, 2), (1, 3), (2, 4), (3, 4), (3, 5), and (3, 6). The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. This is an interactive problem. In the Wonderful Metropolis of the Future, there is no need in subway train drivers. Due to the technological progress, they were replaced by the Artificial Intelligence (AI). Unfortunately, one day the predictions of sci-fi writers came true: the AI rebelled and now there is an uncontrollable train in the subway. It can be dangerous! Your task is to find the train and stop the AI. The subway of the Metropolis is one line (regular straight line with no self-intersections) with n stations, indexed consecutively from 1 to n. At each moment the train is at some station. You need to determine the index of this station, so that the train would be secured. To find the train, dispatcher Sarah gave you a gadget that allows you to select arbitrary numbers l and r (l ≤ r), and then check, whether the train is located on a station with index between l and r, inclusive. Unfortunately, recharging of the gadget takes some time (and every time you use it as soon as possible), so between two applications of the gadget the train can move to any station that is at most k stations away. Formally, if the train was at the station x when the gadget was applied, then at the next application of the gadget the train can appear at any station y such that max(1, x - k) ≤ y ≤ min(n, x + k). Note that AI is not aware that you are trying to catch the train, so it makes all moves according to its predefined plan. After an examination of the gadget you found that it is very old and can hold no more than 4500 applications, after which it will break and your mission will be considered a failure. Can you find the station with the train using no more than 4500 applications of the gadgets? Input The first line contains two integers n and k (1 ≤ n ≤ 10^{18}, 0 ≤ k ≤ 10) — the number of stations and the maximum number of stations the train can move between two applications of the gadget. Interaction You can apply the gadget at most 4500 times. In order to apply the gadget you need to print two space-separated integers l and r (1 ≤ l ≤ r ≤ n). You will then receive either string "Yes", if the train is between stations l and r, inclusive, or string "No" otherwise. If l = r and you received "Yes", then you found the train successfully, and your program must halt immediately. Answer "Bad" instead of "Yes" or "No" means that you made an invalid query or made too many queries. Exit immediately after receiving "Bad" and you will see Wrong answer verdict. Otherwise you can get an arbitrary verdict because your solution will continue to read from a closed stream. After printing a query do not forget to output end of line and flush the output. Otherwise you will get Idleness limit exceeded. To do this, use: * fflush(stdout) or cout.flush() in C++; * System.out.flush() in Java; * flush(output) in Pascal; * stdout.flush() in Python; * see documentation for other languages. Hacks In order to hack, you should present a test in the following format. The first line should contain three integers n, k and p (1 ≤ n ≤ 10^{18}, 0 ≤ k ≤ 10, 1 ≤ p ≤ n) — the number of stations, the maximum number of stations the train can move between two applications of the gadget and the initial position of the train, respectively. Each of the next 4500 lines should contain a single integer x (1 ≤ x ≤ n) — the positions of the train after each query. Two consecutive positions (including the initial one) should not differ by more than k. For example, the following lines are the first lines of the sample test. 10 2 5 5 3 5 7 7 ... Example Input 10 2 Yes No Yes Yes Output 3 5 3 3 3 4 5 5 Note In the first sample, the train was initially at the station 5, after the first application of the gadget it did not move, after the second application it moved to the station 3, and after the third application moved again to the station 5. The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. The Squareland national forest is divided into equal 1 × 1 square plots aligned with north-south and east-west directions. Each plot can be uniquely described by integer Cartesian coordinates (x, y) of its south-west corner. Three friends, Alice, Bob, and Charlie are going to buy three distinct plots of land A, B, C in the forest. Initially, all plots in the forest (including the plots A, B, C) are covered by trees. The friends want to visit each other, so they want to clean some of the plots from trees. After cleaning, one should be able to reach any of the plots A, B, C from any other one of those by moving through adjacent cleared plots. Two plots are adjacent if they share a side. <image> For example, A=(0,0), B=(1,1), C=(2,2). The minimal number of plots to be cleared is 5. One of the ways to do it is shown with the gray color. Of course, the friends don't want to strain too much. Help them find out the smallest number of plots they need to clean from trees. Input The first line contains two integers x_A and y_A — coordinates of the plot A (0 ≤ x_A, y_A ≤ 1000). The following two lines describe coordinates (x_B, y_B) and (x_C, y_C) of plots B and C respectively in the same format (0 ≤ x_B, y_B, x_C, y_C ≤ 1000). It is guaranteed that all three plots are distinct. Output On the first line print a single integer k — the smallest number of plots needed to be cleaned from trees. The following k lines should contain coordinates of all plots needed to be cleaned. All k plots should be distinct. You can output the plots in any order. If there are multiple solutions, print any of them. Examples Input 0 0 1 1 2 2 Output 5 0 0 1 0 1 1 1 2 2 2 Input 0 0 2 0 1 1 Output 4 0 0 1 0 1 1 2 0 Note The first example is shown on the picture in the legend. The second example is illustrated with the following image: <image> The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Salem gave you n sticks with integer positive lengths a_1, a_2, …, a_n. For every stick, you can change its length to any other positive integer length (that is, either shrink or stretch it). The cost of changing the stick's length from a to b is \|a - b\|, where \|x\| means the absolute value of x. A stick length a_i is called almost good for some integer t if \|a_i - t\| ≤ 1. Salem asks you to change the lengths of some sticks (possibly all or none), such that all sticks' lengths are almost good for some positive integer t and the total cost of changing is minimum possible. The value of t is not fixed in advance and you can choose it as any positive integer. As an answer, print the value of t and the minimum cost. If there are multiple optimal choices for t, print any of them. Input The first line contains a single integer n (1 ≤ n ≤ 1000) — the number of sticks. The second line contains n integers a_i (1 ≤ a_i ≤ 100) — the lengths of the sticks. Output Print the value of t and the minimum possible cost. If there are multiple optimal choices for t, print any of them. Examples Input 3 10 1 4 Output 3 7 Input 5 1 1 2 2 3 Output 2 0 Note In the first example, we can change 1 into 2 and 10 into 4 with cost \|1 - 2\| + \|10 - 4\| = 1 + 6 = 7 and the resulting lengths [2, 4, 4] are almost good for t = 3. In the second example, the sticks lengths are already almost good for t = 2, so we don't have to do anything. The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Berland SU holds yet another training contest for its students today. n students came, each of them brought his laptop. However, it turned out that everyone has forgot their chargers! Let students be numbered from 1 to n. Laptop of the i-th student has charge a_i at the beginning of the contest and it uses b_i of charge per minute (i.e. if the laptop has c charge at the beginning of some minute, it becomes c - b_i charge at the beginning of the next minute). The whole contest lasts for k minutes. Polycarp (the coach of Berland SU) decided to buy a single charger so that all the students would be able to successfully finish the contest. He buys the charger at the same moment the contest starts. Polycarp can choose to buy the charger with any non-negative (zero or positive) integer power output. The power output is chosen before the purchase, it can't be changed afterwards. Let the chosen power output be x. At the beginning of each minute (from the minute contest starts to the last minute of the contest) he can plug the charger into any of the student's laptops and use it for some integer number of minutes. If the laptop is using b_i charge per minute then it will become b_i - x per minute while the charger is plugged in. Negative power usage rate means that the laptop's charge is increasing. The charge of any laptop isn't limited, it can become infinitely large. The charger can be plugged in no more than one laptop at the same time. The student successfully finishes the contest if the charge of his laptop never is below zero at the beginning of some minute (from the minute contest starts to the last minute of the contest, zero charge is allowed). The charge of the laptop of the minute the contest ends doesn't matter. Help Polycarp to determine the minimal possible power output the charger should have so that all the students are able to successfully finish the contest. Also report if no such charger exists. Input The first line contains two integers n and k (1 ≤ n ≤ 2 ⋅ 10^5, 1 ≤ k ≤ 2 ⋅ 10^5) — the number of students (and laptops, correspondigly) and the duration of the contest in minutes. The second line contains n integers a_1, a_2, ..., a_n (1 ≤ a_i ≤ 10^{12}) — the initial charge of each student's laptop. The third line contains n integers b_1, b_2, ..., b_n (1 ≤ b_i ≤ 10^7) — the power usage of each student's laptop. Output Print a single non-negative integer — the minimal possible power output the charger should have so that all the students are able to successfully finish the contest. If no such charger exists, print -1. Examples Input 2 4 3 2 4 2 Output 5 Input 1 5 4 2 Output 1 Input 1 6 4 2 Output 2 Input 2 2 2 10 3 15 Output -1 Note Let's take a look at the state of laptops in the beginning of each minute on the first example with the charger of power 5: 1. charge: [3, 2], plug the charger into laptop 1; 2. charge: [3 - 4 + 5, 2 - 2] = [4, 0], plug the charger into laptop 2; 3. charge: [4 - 4, 0 - 2 + 5] = [0, 3], plug the charger into laptop 1; 4. charge: [0 - 4 + 5, 3 - 2] = [1, 1]. The contest ends after the fourth minute. However, let's consider the charger of power 4: 1. charge: [3, 2], plug the charger into laptop 1; 2. charge: [3 - 4 + 4, 2 - 2] = [3, 0], plug the charger into laptop 2; 3. charge: [3 - 4, 0 - 2 + 4] = [-1, 2], the first laptop has negative charge, thus, the first student doesn't finish the contest. In the fourth example no matter how powerful the charger is, one of the students won't finish the contest. The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Let's define a function f(p) on a permutation p as follows. Let g_i be the [greatest common divisor (GCD)](https://en.wikipedia.org/wiki/Greatest_common_divisor) of elements p_1, p_2, ..., p_i (in other words, it is the GCD of the prefix of length i). Then f(p) is the number of distinct elements among g_1, g_2, ..., g_n. Let f_{max}(n) be the maximum value of f(p) among all permutations p of integers 1, 2, ..., n. Given an integers n, count the number of permutations p of integers 1, 2, ..., n, such that f(p) is equal to f_{max}(n). Since the answer may be large, print the remainder of its division by 1000 000 007 = 10^9 + 7. Input The only line contains the integer n (2 ≤ n ≤ 10^6) — the length of the permutations. Output The only line should contain your answer modulo 10^9+7. Examples Input 2 Output 1 Input 3 Output 4 Input 6 Output 120 Note Consider the second example: these are the permutations of length 3: * [1,2,3], f(p)=1. * [1,3,2], f(p)=1. * [2,1,3], f(p)=2. * [2,3,1], f(p)=2. * [3,1,2], f(p)=2. * [3,2,1], f(p)=2. The maximum value f_{max}(3) = 2, and there are 4 permutations p such that f(p)=2. The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Kamil likes streaming the competitive programming videos. His MeTube channel has recently reached 100 million subscribers. In order to celebrate this, he posted a video with an interesting problem he couldn't solve yet. Can you help him? You're given a tree — a connected undirected graph consisting of n vertices connected by n - 1 edges. The tree is rooted at vertex 1. A vertex u is called an ancestor of v if it lies on the shortest path between the root and v. In particular, a vertex is an ancestor of itself. Each vertex v is assigned its beauty x_v — a non-negative integer not larger than 10^{12}. This allows us to define the beauty of a path. Let u be an ancestor of v. Then we define the beauty f(u, v) as the greatest common divisor of the beauties of all vertices on the shortest path between u and v. Formally, if u=t_1, t_2, t_3, ..., t_k=v are the vertices on the shortest path between u and v, then f(u, v) = \gcd(x_{t_1}, x_{t_2}, ..., x_{t_k}). Here, \gcd denotes the greatest common divisor of a set of numbers. In particular, f(u, u) = \gcd(x_u) = x_u. Your task is to find the sum $$$ ∑_{u is an ancestor of v} f(u, v). $$$ As the result might be too large, please output it modulo 10^9 + 7. Note that for each y, \gcd(0, y) = \gcd(y, 0) = y. In particular, \gcd(0, 0) = 0. Input The first line contains a single integer n (2 ≤ n ≤ 100 000) — the number of vertices in the tree. The following line contains n integers x_1, x_2, ..., x_n (0 ≤ x_i ≤ 10^{12}). The value x_v denotes the beauty of vertex v. The following n - 1 lines describe the edges of the tree. Each of them contains two integers a, b (1 ≤ a, b ≤ n, a ≠ b) — the vertices connected by a single edge. Output Output the sum of the beauties on all paths (u, v) such that u is ancestor of v. This sum should be printed modulo 10^9 + 7. Examples Input 5 4 5 6 0 8 1 2 1 3 1 4 4 5 Output 42 Input 7 0 2 3 0 0 0 0 1 2 1 3 2 4 2 5 3 6 3 7 Output 30 Note The following figure shows all 10 possible paths for which one endpoint is an ancestor of another endpoint. The sum of beauties of all these paths is equal to 42: <image> The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Recently, Norge found a string s = s_1 s_2 … s_n consisting of n lowercase Latin letters. As an exercise to improve his typing speed, he decided to type all substrings of the string s. Yes, all (n (n + 1))/(2) of them! A substring of s is a non-empty string x = s[a … b] = s_{a} s_{a + 1} … s_{b} (1 ≤ a ≤ b ≤ n). For example, "auto" and "ton" are substrings of "automaton". Shortly after the start of the exercise, Norge realized that his keyboard was broken, namely, he could use only k Latin letters c_1, c_2, …, c_k out of 26. After that, Norge became interested in how many substrings of the string s he could still type using his broken keyboard. Help him to find this number. Input The first line contains two space-separated integers n and k (1 ≤ n ≤ 2 ⋅ 10^5, 1 ≤ k ≤ 26) — the length of the string s and the number of Latin letters still available on the keyboard. The second line contains the string s consisting of exactly n lowercase Latin letters. The third line contains k space-separated distinct lowercase Latin letters c_1, c_2, …, c_k — the letters still available on the keyboard. Output Print a single number — the number of substrings of s that can be typed using only available letters c_1, c_2, …, c_k. Examples Input 7 2 abacaba a b Output 12 Input 10 3 sadfaasdda f a d Output 21 Input 7 1 aaaaaaa b Output 0 Note In the first example Norge can print substrings s[1…2], s[2…3], s[1…3], s[1…1], s[2…2], s[3…3], s[5…6], s[6…7], s[5…7], s[5…5], s[6…6], s[7…7]. The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. You are given two integers a and m. Calculate the number of integers x such that 0 ≤ x < m and \gcd(a, m) = \gcd(a + x, m). Note: \gcd(a, b) is the greatest common divisor of a and b. Input The first line contains the single integer T (1 ≤ T ≤ 50) — the number of test cases. Next T lines contain test cases — one per line. Each line contains two integers a and m (1 ≤ a < m ≤ 10^{10}). Output Print T integers — one per test case. For each test case print the number of appropriate x-s. Example Input 3 4 9 5 10 42 9999999967 Output 6 1 9999999966 Note In the first test case appropriate x-s are [0, 1, 3, 4, 6, 7]. In the second test case the only appropriate x is 0. The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Nash designed an interesting yet simple board game where a player is simply required to follow instructions written on the cell where the player currently stands. This board game is played on the n× n board. Rows and columns of this board are numbered from 1 to n. The cell on the intersection of the r-th row and c-th column is denoted by (r, c). Some cells on the board are called blocked zones. On each cell of the board, there is written one of the following 5 characters — U, D, L, R or X — instructions for the player. Suppose that the current cell is (r, c). If the character is R, the player should move to the right cell (r, c+1), for L the player should move to the left cell (r, c-1), for U the player should move to the top cell (r-1, c), for D the player should move to the bottom cell (r+1, c). Finally, if the character in the cell is X, then this cell is the blocked zone. The player should remain in this cell (the game for him isn't very interesting from now on). It is guaranteed that the characters are written in a way that the player will never have to step outside of the board, no matter at which cell he starts. As a player starts from a cell, he moves according to the character in the current cell. The player keeps moving until he lands in a blocked zone. It is also possible that the player will keep moving infinitely long. For every of the n^2 cells of the board Alice, your friend, wants to know, how will the game go, if the player starts in this cell. For each starting cell of the board, she writes down the cell that the player stops at, or that the player never stops at all. She gives you the information she has written: for each cell (r, c) she wrote: * a pair (x,y), meaning if a player had started at (r, c), he would end up at cell (x,y). * or a pair (-1,-1), meaning if a player had started at (r, c), he would keep moving infinitely long and would never enter the blocked zone. It might be possible that Alice is trying to fool you and there's no possible grid that satisfies all the constraints Alice gave you. For the given information Alice provided you, you are required to decipher a possible board, or to determine that such a board doesn't exist. If there exist several different boards that satisfy the provided information, you can find any of them. Input The first line of the input contains a single integer n (1 ≤ n ≤ 10^{3}) — the side of the board. The i-th of the next n lines of the input contains 2n integers x_1, y_1, x_2, y_2, ..., x_n, y_n, where (x_j, y_j) (1 ≤ x_j ≤ n, 1 ≤ y_j ≤ n, or (x_j,y_j)=(-1,-1)) is the pair written by Alice for the cell (i, j). Output If there doesn't exist a board satisfying the information that Alice gave you, print a single line containing INVALID. Otherwise, in the first line print VALID. In the i-th of the next n lines, print the string of n characters, corresponding to the characters in the i-th row of the suitable board you found. Each character of a string can either be U, D, L, R or X. If there exist several different boards that satisfy the provided information, you can find any of them. Examples Input 2 1 1 1 1 2 2 2 2 Output VALID XL RX Input 3 -1 -1 -1 -1 -1 -1 -1 -1 2 2 -1 -1 -1 -1 -1 -1 -1 -1 Output VALID RRD UXD ULL Note For the sample test 1 : The given grid in output is a valid one. * If the player starts at (1,1), he doesn't move any further following X and stops there. * If the player starts at (1,2), he moves to left following L and stops at (1,1). * If the player starts at (2,1), he moves to right following R and stops at (2,2). * If the player starts at (2,2), he doesn't move any further following X and stops there. The simulation can be seen below : <image> For the sample test 2 : The given grid in output is a valid one, as a player starting at any cell other than the one at center (2,2), keeps moving in an infinitely long cycle and never stops. Had he started at (2,2), he wouldn't have moved further following instruction X . The simulation can be seen below : <image> The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Having read half of the book called "Storm and Calm" on the IT lesson, Innocentius was absolutely determined to finish the book on the maths lessons. All was fine until the math teacher Ms. Watkins saw Innocentius reading fiction books instead of solving equations of the fifth degree. As during the last maths class Innocentius suggested the algorithm of solving equations of the fifth degree in the general case, Ms. Watkins had no other choice but to give him a new task. The teacher asked to write consecutively (without spaces) all words from the "Storm and Calm" in one long string s. She thought that a string is good if the number of vowels in the string is no more than twice more than the number of consonants. That is, the string with v vowels and c consonants is good if and only if v ≤ 2c. The task Innocentius had to solve turned out to be rather simple: he should find the number of the longest good substrings of the string s. Input The only input line contains a non-empty string s consisting of no more than 2·105 uppercase and lowercase Latin letters. We shall regard letters "a", "e", "i", "o", "u" and their uppercase variants as vowels. Output Print on a single line two numbers without a space: the maximum length of a good substring and the number of good substrings with this length. If no good substring exists, print "No solution" without the quotes. Two substrings are considered different if their positions of occurrence are different. So if some string occurs more than once, then it should be counted more than once. Examples Input Abo Output 3 1 Input OEIS Output 3 1 Input auBAAbeelii Output 9 3 Input AaaBRAaaCAaaDAaaBRAaa Output 18 4 Input EA Output No solution Note In the first sample there is only one longest good substring: "Abo" itself. The other good substrings are "b", "Ab", "bo", but these substrings have shorter length. In the second sample there is only one longest good substring: "EIS". The other good substrings are: "S", "IS". The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Oleg's favorite subjects are History and Math, and his favorite branch of mathematics is division. To improve his division skills, Oleg came up with t pairs of integers p_i and q_i and for each pair decided to find the greatest integer x_i, such that: * p_i is divisible by x_i; * x_i is not divisible by q_i. Oleg is really good at division and managed to find all the answers quickly, how about you? Input The first line contains an integer t (1 ≤ t ≤ 50) — the number of pairs. Each of the following t lines contains two integers p_i and q_i (1 ≤ p_i ≤ 10^{18}; 2 ≤ q_i ≤ 10^{9}) — the i-th pair of integers. Output Print t integers: the i-th integer is the largest x_i such that p_i is divisible by x_i, but x_i is not divisible by q_i. One can show that there is always at least one value of x_i satisfying the divisibility conditions for the given constraints. Example Input 3 10 4 12 6 179 822 Output 10 4 179 Note For the first pair, where p_1 = 10 and q_1 = 4, the answer is x_1 = 10, since it is the greatest divisor of 10 and 10 is not divisible by 4. For the second pair, where p_2 = 12 and q_2 = 6, note that * 12 is not a valid x_2, since 12 is divisible by q_2 = 6; * 6 is not valid x_2 as well: 6 is also divisible by q_2 = 6. The next available divisor of p_2 = 12 is 4, which is the answer, since 4 is not divisible by 6. The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Monocarp had a sequence a consisting of n + m integers a_1, a_2, ..., a_{n + m}. He painted the elements into two colors, red and blue; n elements were painted red, all other m elements were painted blue. After painting the elements, he has written two sequences r_1, r_2, ..., r_n and b_1, b_2, ..., b_m. The sequence r consisted of all red elements of a in the order they appeared in a; similarly, the sequence b consisted of all blue elements of a in the order they appeared in a as well. Unfortunately, the original sequence was lost, and Monocarp only has the sequences r and b. He wants to restore the original sequence. In case there are multiple ways to restore it, he wants to choose a way to restore that maximizes the value of $$$f(a) = max(0, a_1, (a_1 + a_2), (a_1 + a_2 + a_3), ..., (a_1 + a_2 + a_3 + ... + a_{n + m}))$$$ Help Monocarp to calculate the maximum possible value of f(a). Input The first line contains one integer t (1 ≤ t ≤ 1000) — the number of test cases. Then the test cases follow. Each test case consists of four lines. The first line of each test case contains one integer n (1 ≤ n ≤ 100). The second line contains n integers r_1, r_2, ..., r_n (-100 ≤ r_i ≤ 100). The third line contains one integer m (1 ≤ m ≤ 100). The fourth line contains m integers b_1, b_2, ..., b_m (-100 ≤ b_i ≤ 100). Output For each test case, print one integer — the maximum possible value of f(a). Example Input 4 4 6 -5 7 -3 3 2 3 -4 2 1 1 4 10 -3 2 2 5 -1 -2 -3 -4 -5 5 -1 -2 -3 -4 -5 1 0 1 0 Output 13 13 0 0 Note In the explanations for the sample test cases, red elements are marked as bold. In the first test case, one of the possible sequences a is [6, 2, -5, 3, 7, -3, -4]. In the second test case, one of the possible sequences a is [10, 1, -3, 1, 2, 2]. In the third test case, one of the possible sequences a is [-1, -1, -2, -3, -2, -4, -5, -3, -4, -5]. In the fourth test case, one of the possible sequences a is [0, 0]. The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Cirno gave AquaMoon a chessboard of size 1 × n. Its cells are numbered with integers from 1 to n from left to right. In the beginning, some of the cells are occupied with at most one pawn, and other cells are unoccupied. In each operation, AquaMoon can choose a cell i with a pawn, and do either of the following (if possible): * Move pawn from it to the (i+2)-th cell, if i+2 ≤ n and the (i+1)-th cell is occupied and the (i+2)-th cell is unoccupied. * Move pawn from it to the (i-2)-th cell, if i-2 ≥ 1 and the (i-1)-th cell is occupied and the (i-2)-th cell is unoccupied. You are given an initial state of the chessboard. AquaMoon wants to count the number of states reachable from the initial state with some sequence of operations. But she is not good at programming. Can you help her? As the answer can be large find it modulo 998 244 353. Input The input consists of multiple test cases. The first line contains a single integer t (1 ≤ t ≤ 10 000) — the number of test cases. The first line contains a single integer n (1 ≤ n ≤ 10^5) — the size of the chessboard. The second line contains a string of n characters, consists of characters "0" and "1". If the i-th character is "1", the i-th cell is initially occupied; otherwise, the i-th cell is initially unoccupied. It is guaranteed that the sum of n over all test cases does not exceed 10^5. Output For each test case, print the number of states that reachable from the initial state with some sequence of operations modulo 998 244 353. Example Input 6 4 0110 6 011011 5 01010 20 10001111110110111000 20 00110110100110111101 20 11101111011000100010 Output 3 6 1 1287 1287 715 Note In the first test case the strings "1100", "0110" and "0011" are reachable from the initial state with some sequence of operations. The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. In Berland the opposition is going to arrange mass walking on the boulevard. The boulevard consists of n tiles that are lain in a row and are numbered from 1 to n from right to left. The opposition should start walking on the tile number 1 and the finish on the tile number n. During the walk it is allowed to move from right to left between adjacent tiles in a row, and jump over a tile. More formally, if you are standing on the tile number i (i < n - 1), you can reach the tiles number i + 1 or the tile number i + 2 from it (if you stand on the tile number n - 1, you can only reach tile number n). We can assume that all the opposition movements occur instantaneously. In order to thwart an opposition rally, the Berland bloody regime organized the rain. The tiles on the boulevard are of poor quality and they are rapidly destroyed in the rain. We know that the i-th tile is destroyed after ai days of rain (on day ai tile isn't destroyed yet, and on day ai + 1 it is already destroyed). Of course, no one is allowed to walk on the destroyed tiles! So the walk of the opposition is considered thwarted, if either the tile number 1 is broken, or the tile number n is broken, or it is impossible to reach the tile number n from the tile number 1 if we can walk on undestroyed tiles. The opposition wants to gather more supporters for their walk. Therefore, the more time they have to pack, the better. Help the opposition to calculate how much time they still have and tell us for how many days the walk from the tile number 1 to the tile number n will be possible. Input The first line contains integer n (1 ≤ n ≤ 103) — the boulevard's length in tiles. The second line contains n space-separated integers ai — the number of days after which the i-th tile gets destroyed (1 ≤ ai ≤ 103). Output Print a single number — the sought number of days. Examples Input 4 10 3 5 10 Output 5 Input 5 10 2 8 3 5 Output 5 Note In the first sample the second tile gets destroyed after day three, and the only path left is 1 → 3 → 4. After day five there is a two-tile gap between the first and the last tile, you can't jump over it. In the second sample path 1 → 3 → 5 is available up to day five, inclusive. On day six the last tile is destroyed and the walk is thwarted. The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. One day n students come to the stadium. They want to play football, and for that they need to split into teams, the teams must have an equal number of people. We know that this group of people has archenemies. Each student has at most two archenemies. Besides, if student A is an archenemy to student B, then student B is an archenemy to student A. The students want to split so as no two archenemies were in one team. If splitting in the required manner is impossible, some students will have to sit on the bench. Determine the minimum number of students you will have to send to the bench in order to form the two teams in the described manner and begin the game at last. Input The first line contains two integers n and m (2 ≤ n ≤ 100, 1 ≤ m ≤ 100) — the number of students and the number of pairs of archenemies correspondingly. Next m lines describe enmity between students. Each enmity is described as two numbers ai and bi (1 ≤ ai, bi ≤ n, ai ≠ bi) — the indexes of the students who are enemies to each other. Each enmity occurs in the list exactly once. It is guaranteed that each student has no more than two archenemies. You can consider the students indexed in some manner with distinct integers from 1 to n. Output Print a single integer — the minimum number of students you will have to send to the bench in order to start the game. Examples Input 5 4 1 2 2 4 5 3 1 4 Output 1 Input 6 2 1 4 3 4 Output 0 Input 6 6 1 2 2 3 3 1 4 5 5 6 6 4 Output 2 The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. n people came to a party. Then those, who had no friends among people at the party, left. Then those, who had exactly 1 friend among those who stayed, left as well. Then those, who had exactly 2, 3, ..., n - 1 friends among those who stayed by the moment of their leaving, did the same. What is the maximum amount of people that could stay at the party in the end? Input The first input line contains one number t — amount of tests (1 ≤ t ≤ 105). Each of the following t lines contains one integer number n (1 ≤ n ≤ 105). Output For each test output in a separate line one number — the maximum amount of people that could stay in the end. Examples Input 1 3 Output 1 The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. One beautiful day Vasily the bear painted 2m circles of the same radius R on a coordinate plane. Circles with numbers from 1 to m had centers at points (2R - R, 0), (4R - R, 0), ..., (2Rm - R, 0), respectively. Circles with numbers from m + 1 to 2m had centers at points (2R - R, 2R), (4R - R, 2R), ..., (2Rm - R, 2R), respectively. Naturally, the bear painted the circles for a simple experiment with a fly. The experiment continued for m2 days. Each day of the experiment got its own unique number from 0 to m2 - 1, inclusive. On the day number i the following things happened: 1. The fly arrived at the coordinate plane at the center of the circle with number <image> (<image> is the result of dividing number x by number y, rounded down to an integer). 2. The fly went along the coordinate plane to the center of the circle number <image> (<image> is the remainder after dividing number x by number y). The bear noticed that the fly went from the center of circle v to the center of circle u along the shortest path with all points lying on the border or inside at least one of the 2m circles. After the fly reached the center of circle u, it flew away in an unknown direction. Help Vasily, count the average distance the fly went along the coordinate plane during each of these m2 days. Input The first line contains two integers m, R (1 ≤ m ≤ 105, 1 ≤ R ≤ 10). Output In a single line print a single real number — the answer to the problem. The answer will be considered correct if its absolute or relative error doesn't exceed 10 - 6. Examples Input 1 1 Output 2.0000000000 Input 2 2 Output 5.4142135624 Note <image> Figure to the second sample The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Simon has a prime number x and an array of non-negative integers a1, a2, ..., an. Simon loves fractions very much. Today he wrote out number <image> on a piece of paper. After Simon led all fractions to a common denominator and summed them up, he got a fraction: <image>, where number t equals xa1 + a2 + ... + an. Now Simon wants to reduce the resulting fraction. Help him, find the greatest common divisor of numbers s and t. As GCD can be rather large, print it as a remainder after dividing it by number 1000000007 (109 + 7). Input The first line contains two positive integers n and x (1 ≤ n ≤ 105, 2 ≤ x ≤ 109) — the size of the array and the prime number. The second line contains n space-separated integers a1, a2, ..., an (0 ≤ a1 ≤ a2 ≤ ... ≤ an ≤ 109). Output Print a single number — the answer to the problem modulo 1000000007 (109 + 7). Examples Input 2 2 2 2 Output 8 Input 3 3 1 2 3 Output 27 Input 2 2 29 29 Output 73741817 Input 4 5 0 0 0 0 Output 1 Note In the first sample <image>. Thus, the answer to the problem is 8. In the second sample, <image>. The answer to the problem is 27, as 351 = 13·27, 729 = 27·27. In the third sample the answer to the problem is 1073741824 mod 1000000007 = 73741817. In the fourth sample <image>. Thus, the answer to the problem is 1. The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Arthur and Alexander are number busters. Today they've got a competition. Arthur took a group of four integers a, b, w, x (0 ≤ b < w, 0 < x < w) and Alexander took integer с. Arthur and Alexander use distinct approaches to number bustings. Alexander is just a regular guy. Each second, he subtracts one from his number. In other words, he performs the assignment: c = c - 1. Arthur is a sophisticated guy. Each second Arthur performs a complex operation, described as follows: if b ≥ x, perform the assignment b = b - x, if b < x, then perform two consecutive assignments a = a - 1; b = w - (x - b). You've got numbers a, b, w, x, c. Determine when Alexander gets ahead of Arthur if both guys start performing the operations at the same time. Assume that Alexander got ahead of Arthur if c ≤ a. Input The first line contains integers a, b, w, x, c (1 ≤ a ≤ 2·109, 1 ≤ w ≤ 1000, 0 ≤ b < w, 0 < x < w, 1 ≤ c ≤ 2·109). Output Print a single integer — the minimum time in seconds Alexander needs to get ahead of Arthur. You can prove that the described situation always occurs within the problem's limits. Examples Input 4 2 3 1 6 Output 2 Input 4 2 3 1 7 Output 4 Input 1 2 3 2 6 Output 13 Input 1 1 2 1 1 Output 0 The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Let's call an undirected graph of n vertices p-interesting, if the following conditions fulfill: * the graph contains exactly 2n + p edges; * the graph doesn't contain self-loops and multiple edges; * for any integer k (1 ≤ k ≤ n), any subgraph consisting of k vertices contains at most 2k + p edges. A subgraph of a graph is some set of the graph vertices and some set of the graph edges. At that, the set of edges must meet the condition: both ends of each edge from the set must belong to the chosen set of vertices. Your task is to find a p-interesting graph consisting of n vertices. Input The first line contains a single integer t (1 ≤ t ≤ 5) — the number of tests in the input. Next t lines each contains two space-separated integers: n, p (5 ≤ n ≤ 24; p ≥ 0; <image>) — the number of vertices in the graph and the interest value for the appropriate test. It is guaranteed that the required graph exists. Output For each of the t tests print 2n + p lines containing the description of the edges of a p-interesting graph: the i-th line must contain two space-separated integers ai, bi (1 ≤ ai, bi ≤ n; ai ≠ bi) — two vertices, connected by an edge in the resulting graph. Consider the graph vertices numbered with integers from 1 to n. Print the answers to the tests in the order the tests occur in the input. If there are multiple solutions, you can print any of them. Examples Input 1 6 0 Output 1 2 1 3 1 4 1 5 1 6 2 3 2 4 2 5 2 6 3 4 3 5 3 6 The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Iahub is training for the IOI. What is a better way to train than playing a Zuma-like game? There are n balls put in a row. Each ball is colored in one of k colors. Initially the row doesn't contain three or more contiguous balls with the same color. Iahub has a single ball of color x. He can insert his ball at any position in the row (probably, between two other balls). If at any moment there are three or more contiguous balls of the same color in the row, they are destroyed immediately. This rule is applied multiple times, until there are no more sets of 3 or more contiguous balls of the same color. For example, if Iahub has the row of balls [black, black, white, white, black, black] and a white ball, he can insert the ball between two white balls. Thus three white balls are destroyed, and then four black balls become contiguous, so all four balls are destroyed. The row will not contain any ball in the end, so Iahub can destroy all 6 balls. Iahub wants to destroy as many balls as possible. You are given the description of the row of balls, and the color of Iahub's ball. Help Iahub train for the IOI by telling him the maximum number of balls from the row he can destroy. Input The first line of input contains three integers: n (1 ≤ n ≤ 100), k (1 ≤ k ≤ 100) and x (1 ≤ x ≤ k). The next line contains n space-separated integers c1, c2, ..., cn (1 ≤ ci ≤ k). Number ci means that the i-th ball in the row has color ci. It is guaranteed that the initial row of balls will never contain three or more contiguous balls of the same color. Output Print a single integer — the maximum number of balls Iahub can destroy. Examples Input 6 2 2 1 1 2 2 1 1 Output 6 Input 1 1 1 1 Output 0 The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. You are given three strings (s1, s2, s3). For each integer l (1 ≤ l ≤ min(\|s1\|, \|s2\|, \|s3\|) you need to find how many triples (i1, i2, i3) exist such that three strings sk[ik... ik + l - 1] (k = 1, 2, 3) are pairwise equal. Print all found numbers modulo 1000000007 (109 + 7). See notes if you are not sure about some of the denotions used in the statement. Input First three lines contain three non-empty input strings. The sum of lengths of all strings is no more than 3·105. All strings consist only of lowercase English letters. Output You need to output min(\|s1\|, \|s2\|, \|s3\|) numbers separated by spaces — answers for the problem modulo 1000000007 (109 + 7). Examples Input abc bc cbc Output 3 1 Input abacaba abac abcd Output 11 2 0 0 Note Consider a string t = t1t2... t\|t\|, where ti denotes the i-th character of the string, and \|t\| denotes the length of the string. Then t[i... j] (1 ≤ i ≤ j ≤ \|t\|) represents the string titi + 1... tj (substring of t from position i to position j inclusive). The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. You are given sequence a1, a2, ..., an and m queries lj, rj (1 ≤ lj ≤ rj ≤ n). For each query you need to print the minimum distance between such pair of elements ax and ay (x ≠ y), that: * both indexes of the elements lie within range [lj, rj], that is, lj ≤ x, y ≤ rj; * the values of the elements are equal, that is ax = ay. The text above understands distance as \|x - y\|. Input The first line of the input contains a pair of integers n, m (1 ≤ n, m ≤ 5·105) — the length of the sequence and the number of queries, correspondingly. The second line contains the sequence of integers a1, a2, ..., an ( - 109 ≤ ai ≤ 109). Next m lines contain the queries, one per line. Each query is given by a pair of numbers lj, rj (1 ≤ lj ≤ rj ≤ n) — the indexes of the query range limits. Output Print m integers — the answers to each query. If there is no valid match for some query, please print -1 as an answer to this query. Examples Input 5 3 1 1 2 3 2 1 5 2 4 3 5 Output 1 -1 2 Input 6 5 1 2 1 3 2 3 4 6 1 3 2 5 2 4 1 6 Output 2 2 3 -1 2 The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Sasha and Ira are two best friends. But they aren’t just friends, they are software engineers and experts in artificial intelligence. They are developing an algorithm for two bots playing a two-player game. The game is cooperative and turn based. In each turn, one of the players makes a move (it doesn’t matter which player, it's possible that players turns do not alternate). Algorithm for bots that Sasha and Ira are developing works by keeping track of the state the game is in. Each time either bot makes a move, the state changes. And, since the game is very dynamic, it will never go back to the state it was already in at any point in the past. Sasha and Ira are perfectionists and want their algorithm to have an optimal winning strategy. They have noticed that in the optimal winning strategy, both bots make exactly N moves each. But, in order to find the optimal strategy, their algorithm needs to analyze all possible states of the game (they haven’t learned about alpha-beta pruning yet) and pick the best sequence of moves. They are worried about the efficiency of their algorithm and are wondering what is the total number of states of the game that need to be analyzed? Input The first and only line contains integer N. * 1 ≤ N ≤ 106 Output Output should contain a single integer – number of possible states modulo 109 + 7. Examples Input 2 Output 19 Note Start: Game is in state A. * Turn 1: Either bot can make a move (first bot is red and second bot is blue), so there are two possible states after the first turn – B and C. * Turn 2: In both states B and C, either bot can again make a turn, so the list of possible states is expanded to include D, E, F and G. * Turn 3: Red bot already did N=2 moves when in state D, so it cannot make any more moves there. It can make moves when in state E, F and G, so states I, K and M are added to the list. Similarly, blue bot cannot make a move when in state G, but can when in D, E and F, so states H, J and L are added. * Turn 4: Red bot already did N=2 moves when in states H, I and K, so it can only make moves when in J, L and M, so states P, R and S are added. Blue bot cannot make a move when in states J, L and M, but only when in H, I and K, so states N, O and Q are added. Overall, there are 19 possible states of the game their algorithm needs to analyze. <image> The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Shapur was an extremely gifted student. He was great at everything including Combinatorics, Algebra, Number Theory, Geometry, Calculus, etc. He was not only smart but extraordinarily fast! He could manage to sum 1018 numbers in a single second. One day in 230 AD Shapur was trying to find out if any one can possibly do calculations faster than him. As a result he made a very great contest and asked every one to come and take part. In his contest he gave the contestants many different pairs of numbers. Each number is made from digits 0 or 1. The contestants should write a new number corresponding to the given pair of numbers. The rule is simple: The i-th digit of the answer is 1 if and only if the i-th digit of the two given numbers differ. In the other case the i-th digit of the answer is 0. Shapur made many numbers and first tried his own speed. He saw that he can perform these operations on numbers of length ∞ (length of a number is number of digits in it) in a glance! He always gives correct answers so he expects the contestants to give correct answers, too. He is a good fellow so he won't give anyone very big numbers and he always gives one person numbers of same length. Now you are going to take part in Shapur's contest. See if you are faster and more accurate. Input There are two lines in each input. Each of them contains a single number. It is guaranteed that the numbers are made from 0 and 1 only and that their length is same. The numbers may start with 0. The length of each number doesn't exceed 100. Output Write one line — the corresponding answer. Do not omit the leading 0s. Examples Input 1010100 0100101 Output 1110001 Input 000 111 Output 111 Input 1110 1010 Output 0100 Input 01110 01100 Output 00010 The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. First-rate specialists graduate from Berland State Institute of Peace and Friendship. You are one of the most talented students in this university. The education is not easy because you need to have fundamental knowledge in different areas, which sometimes are not related to each other. For example, you should know linguistics very well. You learn a structure of Reberland language as foreign language. In this language words are constructed according to the following rules. First you need to choose the "root" of the word — some string which has more than 4 letters. Then several strings with the length 2 or 3 symbols are appended to this word. The only restriction — it is not allowed to append the same string twice in a row. All these strings are considered to be suffixes of the word (this time we use word "suffix" to describe a morpheme but not the few last characters of the string as you may used to). Here is one exercise that you have found in your task list. You are given the word s. Find all distinct strings with the length 2 or 3, which can be suffixes of this word according to the word constructing rules in Reberland language. Two strings are considered distinct if they have different length or there is a position in which corresponding characters do not match. Let's look at the example: the word abacabaca is given. This word can be obtained in the following ways: <image>, where the root of the word is overlined, and suffixes are marked by "corners". Thus, the set of possible suffixes for this word is {aca, ba, ca}. Input The only line contains a string s (5 ≤ \|s\| ≤ 104) consisting of lowercase English letters. Output On the first line print integer k — a number of distinct possible suffixes. On the next k lines print suffixes. Print suffixes in lexicographical (alphabetical) order. Examples Input abacabaca Output 3 aca ba ca Input abaca Output 0 Note The first test was analysed in the problem statement. In the second example the length of the string equals 5. The length of the root equals 5, so no string can be used as a suffix. The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. One particularly well-known fact about zombies is that they move and think terribly slowly. While we still don't know why their movements are so sluggish, the problem of laggy thinking has been recently resolved. It turns out that the reason is not (as previously suspected) any kind of brain defect – it's the opposite! Independent researchers confirmed that the nervous system of a zombie is highly complicated – it consists of n brains (much like a cow has several stomachs). They are interconnected by brain connectors, which are veins capable of transmitting thoughts between brains. There are two important properties such a brain network should have to function properly: 1. It should be possible to exchange thoughts between any two pairs of brains (perhaps indirectly, through other brains). 2. There should be no redundant brain connectors, that is, removing any brain connector would make property 1 false. If both properties are satisfied, we say that the nervous system is valid. Unfortunately (?), if the system is not valid, the zombie stops thinking and becomes (even more) dead. Your task is to analyze a given nervous system of a zombie and find out whether it is valid. Input The first line of the input contains two space-separated integers n and m (1 ≤ n, m ≤ 1000) denoting the number of brains (which are conveniently numbered from 1 to n) and the number of brain connectors in the nervous system, respectively. In the next m lines, descriptions of brain connectors follow. Every connector is given as a pair of brains a b it connects (1 ≤ a, b ≤ n, a ≠ b). Output The output consists of one line, containing either yes or no depending on whether the nervous system is valid. Examples Input 4 4 1 2 2 3 3 1 4 1 Output no Input 6 5 1 2 2 3 3 4 4 5 3 6 Output yes The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Anton is playing a very interesting computer game, but now he is stuck at one of the levels. To pass to the next level he has to prepare n potions. Anton has a special kettle, that can prepare one potions in x seconds. Also, he knows spells of two types that can faster the process of preparing potions. 1. Spells of this type speed up the preparation time of one potion. There are m spells of this type, the i-th of them costs bi manapoints and changes the preparation time of each potion to ai instead of x. 2. Spells of this type immediately prepare some number of potions. There are k such spells, the i-th of them costs di manapoints and instantly create ci potions. Anton can use no more than one spell of the first type and no more than one spell of the second type, and the total number of manapoints spent should not exceed s. Consider that all spells are used instantly and right before Anton starts to prepare potions. Anton wants to get to the next level as fast as possible, so he is interested in the minimum number of time he needs to spent in order to prepare at least n potions. Input The first line of the input contains three integers n, m, k (1 ≤ n ≤ 2·109, 1 ≤ m, k ≤ 2·105) — the number of potions, Anton has to make, the number of spells of the first type and the number of spells of the second type. The second line of the input contains two integers x and s (2 ≤ x ≤ 2·109, 1 ≤ s ≤ 2·109) — the initial number of seconds required to prepare one potion and the number of manapoints Anton can use. The third line contains m integers ai (1 ≤ ai < x) — the number of seconds it will take to prepare one potion if the i-th spell of the first type is used. The fourth line contains m integers bi (1 ≤ bi ≤ 2·109) — the number of manapoints to use the i-th spell of the first type. There are k integers ci (1 ≤ ci ≤ n) in the fifth line — the number of potions that will be immediately created if the i-th spell of the second type is used. It's guaranteed that ci are not decreasing, i.e. ci ≤ cj if i < j. The sixth line contains k integers di (1 ≤ di ≤ 2·109) — the number of manapoints required to use the i-th spell of the second type. It's guaranteed that di are not decreasing, i.e. di ≤ dj if i < j. Output Print one integer — the minimum time one has to spent in order to prepare n potions. Examples Input 20 3 2 10 99 2 4 3 20 10 40 4 15 10 80 Output 20 Input 20 3 2 10 99 2 4 3 200 100 400 4 15 100 800 Output 200 Note In the first sample, the optimum answer is to use the second spell of the first type that costs 10 manapoints. Thus, the preparation time of each potion changes to 4 seconds. Also, Anton should use the second spell of the second type to instantly prepare 15 potions spending 80 manapoints. The total number of manapoints used is 10 + 80 = 90, and the preparation time is 4·5 = 20 seconds (15 potions were prepared instantly, and the remaining 5 will take 4 seconds each). In the second sample, Anton can't use any of the spells, so he just prepares 20 potions, spending 10 seconds on each of them and the answer is 20·10 = 200. The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. On the Literature lesson Sergei noticed an awful injustice, it seems that some students are asked more often than others. Seating in the class looks like a rectangle, where n rows with m pupils in each. The teacher asks pupils in the following order: at first, she asks all pupils from the first row in the order of their seating, then she continues to ask pupils from the next row. If the teacher asked the last row, then the direction of the poll changes, it means that she asks the previous row. The order of asking the rows looks as follows: the 1-st row, the 2-nd row, ..., the n - 1-st row, the n-th row, the n - 1-st row, ..., the 2-nd row, the 1-st row, the 2-nd row, ... The order of asking of pupils on the same row is always the same: the 1-st pupil, the 2-nd pupil, ..., the m-th pupil. During the lesson the teacher managed to ask exactly k questions from pupils in order described above. Sergei seats on the x-th row, on the y-th place in the row. Sergei decided to prove to the teacher that pupils are asked irregularly, help him count three values: 1. the maximum number of questions a particular pupil is asked, 2. the minimum number of questions a particular pupil is asked, 3. how many times the teacher asked Sergei. If there is only one row in the class, then the teacher always asks children from this row. Input The first and the only line contains five integers n, m, k, x and y (1 ≤ n, m ≤ 100, 1 ≤ k ≤ 1018, 1 ≤ x ≤ n, 1 ≤ y ≤ m). Output Print three integers: 1. the maximum number of questions a particular pupil is asked, 2. the minimum number of questions a particular pupil is asked, 3. how many times the teacher asked Sergei. Examples Input 1 3 8 1 1 Output 3 2 3 Input 4 2 9 4 2 Output 2 1 1 Input 5 5 25 4 3 Output 1 1 1 Input 100 100 1000000000000000000 100 100 Output 101010101010101 50505050505051 50505050505051 Note The order of asking pupils in the first test: 1. the pupil from the first row who seats at the first table, it means it is Sergei; 2. the pupil from the first row who seats at the second table; 3. the pupil from the first row who seats at the third table; 4. the pupil from the first row who seats at the first table, it means it is Sergei; 5. the pupil from the first row who seats at the second table; 6. the pupil from the first row who seats at the third table; 7. the pupil from the first row who seats at the first table, it means it is Sergei; 8. the pupil from the first row who seats at the second table; The order of asking pupils in the second test: 1. the pupil from the first row who seats at the first table; 2. the pupil from the first row who seats at the second table; 3. the pupil from the second row who seats at the first table; 4. the pupil from the second row who seats at the second table; 5. the pupil from the third row who seats at the first table; 6. the pupil from the third row who seats at the second table; 7. the pupil from the fourth row who seats at the first table; 8. the pupil from the fourth row who seats at the second table, it means it is Sergei; 9. the pupil from the third row who seats at the first table; The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. The year of 2012 is coming... According to an ancient choradrican legend in this very year, in 2012, Diablo and his brothers Mephisto and Baal will escape from hell, and innumerable hordes of demons will enslave the human world. But seven brave heroes have already gathered on the top of a mountain Arreat to protect us mere mortals from the effect of this terrible evil. The seven great heroes are: amazon Anka, barbarian Chapay, sorceress Cleo, druid Troll, necromancer Dracul, paladin Snowy and a professional hit girl Hexadecimal. Heroes already know how much experience will be given for each of the three megabosses: a for Mephisto, b for Diablo and c for Baal. Here's the problem: heroes are as much as seven and megabosses are only three! Then our heroes decided to split into three teams, where each team will go to destroy their own megaboss. Each team member will receive a <image> of experience, rounded down, where x will be the amount of experience for the killed megaboss and y — the number of people in the team. Heroes do not want to hurt each other's feelings, so they want to split into teams so that the difference between the hero who received the maximum number of experience and the hero who received the minimum number of experience were minimal. Since there can be several divisions into teams, then you need to find the one in which the total amount of liking in teams were maximum. It is known that some heroes like others. But if hero p likes hero q, this does not mean that the hero q likes hero p. No hero likes himself. The total amount of liking in teams is the amount of ordered pairs (p, q), such that heroes p and q are in the same group, and hero p likes hero q (but it is not important if hero q likes hero p). In case of heroes p and q likes each other and they are in the same group, this pair should be counted twice, as (p, q) and (q, p). A team can consist even of a single hero, but it is important that every megaboss was destroyed. All heroes must be involved in the campaign against evil. None of the heroes can be in more than one team. It is guaranteed that every hero is able to destroy any megaboss alone. Input The first line contains a single non-negative integer n (0 ≤ n ≤ 42) — amount of liking between the heroes. Next n lines describe liking in the form "p likes q", meaning that the hero p likes the hero q (p ≠ q). Every liking is described in the input exactly once, no hero likes himself. In the last line are given three integers a, b and c (1 ≤ a, b, c ≤ 2·109), separated by spaces: the experience for Mephisto, the experience for Diablo and experience for Baal. In all the pretests, except for examples from the statement, the following condition is satisfied: a = b = c. Output Print two integers — the minimal difference in the experience between two heroes who will receive the maximum and minimum number of experience points, and the maximal total amount of liking in teams (the number of friendships between heroes that end up in one team). When calculating the second answer, the team division should satisfy the difference-minimizing contraint. I.e. primary you should minimize the difference in the experience and secondary you should maximize the total amount of liking. Examples Input 3 Troll likes Dracul Dracul likes Anka Snowy likes Hexadecimal 210 200 180 Output 30 3 Input 2 Anka likes Chapay Chapay likes Anka 10000 50 50 Output 1950 2 Note A note to first example: it the first team should be Dracul, Troll and Anka, in the second one Hexadecimal and Snowy, and in the third Cleo и Chapay. The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Wherever the destination is, whoever we meet, let's render this song together. On a Cartesian coordinate plane lies a rectangular stage of size w × h, represented by a rectangle with corners (0, 0), (w, 0), (w, h) and (0, h). It can be seen that no collisions will happen before one enters the stage. On the sides of the stage stand n dancers. The i-th of them falls into one of the following groups: * Vertical: stands at (xi, 0), moves in positive y direction (upwards); * Horizontal: stands at (0, yi), moves in positive x direction (rightwards). <image> According to choreography, the i-th dancer should stand still for the first ti milliseconds, and then start moving in the specified direction at 1 unit per millisecond, until another border is reached. It is guaranteed that no two dancers have the same group, position and waiting time at the same time. When two dancers collide (i.e. are on the same point at some time when both of them are moving), they immediately exchange their moving directions and go on. <image> Dancers stop when a border of the stage is reached. Find out every dancer's stopping position. Input The first line of input contains three space-separated positive integers n, w and h (1 ≤ n ≤ 100 000, 2 ≤ w, h ≤ 100 000) — the number of dancers and the width and height of the stage, respectively. The following n lines each describes a dancer: the i-th among them contains three space-separated integers gi, pi, and ti (1 ≤ gi ≤ 2, 1 ≤ pi ≤ 99 999, 0 ≤ ti ≤ 100 000), describing a dancer's group gi (gi = 1 — vertical, gi = 2 — horizontal), position, and waiting time. If gi = 1 then pi = xi; otherwise pi = yi. It's guaranteed that 1 ≤ xi ≤ w - 1 and 1 ≤ yi ≤ h - 1. It is guaranteed that no two dancers have the same group, position and waiting time at the same time. Output Output n lines, the i-th of which contains two space-separated integers (xi, yi) — the stopping position of the i-th dancer in the input. Examples Input 8 10 8 1 1 10 1 4 13 1 7 1 1 8 2 2 2 0 2 5 14 2 6 0 2 6 1 Output 4 8 10 5 8 8 10 6 10 2 1 8 7 8 10 6 Input 3 2 3 1 1 2 2 1 1 1 1 5 Output 1 3 2 1 1 3 Note The first example corresponds to the initial setup in the legend, and the tracks of dancers are marked with different colours in the following figure. <image> In the second example, no dancers collide. The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. You are given several queries. In the i-th query you are given a single positive integer ni. You are to represent ni as a sum of maximum possible number of composite summands and print this maximum number, or print -1, if there are no such splittings. An integer greater than 1 is composite, if it is not prime, i.e. if it has positive divisors not equal to 1 and the integer itself. Input The first line contains single integer q (1 ≤ q ≤ 105) — the number of queries. q lines follow. The (i + 1)-th line contains single integer ni (1 ≤ ni ≤ 109) — the i-th query. Output For each query print the maximum possible number of summands in a valid splitting to composite summands, or -1, if there are no such splittings. Examples Input 1 12 Output 3 Input 2 6 8 Output 1 2 Input 3 1 2 3 Output -1 -1 -1 Note 12 = 4 + 4 + 4 = 4 + 8 = 6 + 6 = 12, but the first splitting has the maximum possible number of summands. 8 = 4 + 4, 6 can't be split into several composite summands. 1, 2, 3 are less than any composite number, so they do not have valid splittings. The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Vasya has a non-negative integer n. He wants to round it to nearest integer, which ends up with 0. If n already ends up with 0, Vasya considers it already rounded. For example, if n = 4722 answer is 4720. If n = 5 Vasya can round it to 0 or to 10. Both ways are correct. For given n find out to which integer will Vasya round it. Input The first line contains single integer n (0 ≤ n ≤ 109) — number that Vasya has. Output Print result of rounding n. Pay attention that in some cases answer isn't unique. In that case print any correct answer. Examples Input 5 Output 0 Input 113 Output 110 Input 1000000000 Output 1000000000 Input 5432359 Output 5432360 Note In the first example n = 5. Nearest integers, that ends up with zero are 0 and 10. Any of these answers is correct, so you can print 0 or 10. The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. We consider a positive integer perfect, if and only if the sum of its digits is exactly 10. Given a positive integer k, your task is to find the k-th smallest perfect positive integer. Input A single line with a positive integer k (1 ≤ k ≤ 10 000). Output A single number, denoting the k-th smallest perfect integer. Examples Input 1 Output 19 Input 2 Output 28 Note The first perfect integer is 19 and the second one is 28. The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. You are given a sequence a consisting of n integers. You may partition this sequence into two sequences b and c in such a way that every element belongs exactly to one of these sequences. Let B be the sum of elements belonging to b, and C be the sum of elements belonging to c (if some of these sequences is empty, then its sum is 0). What is the maximum possible value of B - C? Input The first line contains one integer n (1 ≤ n ≤ 100) — the number of elements in a. The second line contains n integers a1, a2, ..., an ( - 100 ≤ ai ≤ 100) — the elements of sequence a. Output Print the maximum possible value of B - C, where B is the sum of elements of sequence b, and C is the sum of elements of sequence c. Examples Input 3 1 -2 0 Output 3 Input 6 16 23 16 15 42 8 Output 120 Note In the first example we may choose b = {1, 0}, c = { - 2}. Then B = 1, C = - 2, B - C = 3. In the second example we choose b = {16, 23, 16, 15, 42, 8}, c = {} (an empty sequence). Then B = 120, C = 0, B - C = 120. The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Petya loves volleyball very much. One day he was running late for a volleyball match. Petya hasn't bought his own car yet, that's why he had to take a taxi. The city has n junctions, some of which are connected by two-way roads. The length of each road is defined by some positive integer number of meters; the roads can have different lengths. Initially each junction has exactly one taxi standing there. The taxi driver from the i-th junction agrees to drive Petya (perhaps through several intermediate junctions) to some other junction if the travel distance is not more than ti meters. Also, the cost of the ride doesn't depend on the distance and is equal to ci bourles. Taxis can't stop in the middle of a road. Each taxi can be used no more than once. Petya can catch taxi only in the junction, where it stands initially. At the moment Petya is located on the junction x and the volleyball stadium is on the junction y. Determine the minimum amount of money Petya will need to drive to the stadium. Input The first line contains two integers n and m (1 ≤ n ≤ 1000, 0 ≤ m ≤ 1000). They are the number of junctions and roads in the city correspondingly. The junctions are numbered from 1 to n, inclusive. The next line contains two integers x and y (1 ≤ x, y ≤ n). They are the numbers of the initial and final junctions correspondingly. Next m lines contain the roads' description. Each road is described by a group of three integers ui, vi, wi (1 ≤ ui, vi ≤ n, 1 ≤ wi ≤ 109) — they are the numbers of the junctions connected by the road and the length of the road, correspondingly. The next n lines contain n pairs of integers ti and ci (1 ≤ ti, ci ≤ 109), which describe the taxi driver that waits at the i-th junction — the maximum distance he can drive and the drive's cost. The road can't connect the junction with itself, but between a pair of junctions there can be more than one road. All consecutive numbers in each line are separated by exactly one space character. Output If taxis can't drive Petya to the destination point, print "-1" (without the quotes). Otherwise, print the drive's minimum cost. Please do not use the %lld specificator to read or write 64-bit integers in С++. It is preferred to use cin, cout streams or the %I64d specificator. Examples Input 4 4 1 3 1 2 3 1 4 1 2 4 1 2 3 5 2 7 7 2 1 2 7 7 Output 9 Note An optimal way — ride from the junction 1 to 2 (via junction 4), then from 2 to 3. It costs 7+2=9 bourles. The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Two participants are each given a pair of distinct numbers from 1 to 9 such that there's exactly one number that is present in both pairs. They want to figure out the number that matches by using a communication channel you have access to without revealing it to you. Both participants communicated to each other a set of pairs of numbers, that includes the pair given to them. Each pair in the communicated sets comprises two different numbers. Determine if you can with certainty deduce the common number, or if you can determine with certainty that both participants know the number but you do not. Input The first line contains two integers n and m (1 ≤ n, m ≤ 12) — the number of pairs the first participant communicated to the second and vice versa. The second line contains n pairs of integers, each between 1 and 9, — pairs of numbers communicated from first participant to the second. The third line contains m pairs of integers, each between 1 and 9, — pairs of numbers communicated from the second participant to the first. All pairs within each set are distinct (in particular, if there is a pair (1,2), there will be no pair (2,1) within the same set), and no pair contains the same number twice. It is guaranteed that the two sets do not contradict the statements, in other words, there is pair from the first set and a pair from the second set that share exactly one number. Output If you can deduce the shared number with certainty, print that number. If you can with certainty deduce that both participants know the shared number, but you do not know it, print 0. Otherwise print -1. Examples Input 2 2 1 2 3 4 1 5 3 4 Output 1 Input 2 2 1 2 3 4 1 5 6 4 Output 0 Input 2 3 1 2 4 5 1 2 1 3 2 3 Output -1 Note In the first example the first participant communicated pairs (1,2) and (3,4), and the second communicated (1,5), (3,4). Since we know that the actual pairs they received share exactly one number, it can't be that they both have (3,4). Thus, the first participant has (1,2) and the second has (1,5), and at this point you already know the shared number is 1. In the second example either the first participant has (1,2) and the second has (1,5), or the first has (3,4) and the second has (6,4). In the first case both of them know the shared number is 1, in the second case both of them know the shared number is 4. You don't have enough information to tell 1 and 4 apart. In the third case if the first participant was given (1,2), they don't know what the shared number is, since from their perspective the second participant might have been given either (1,3), in which case the shared number is 1, or (2,3), in which case the shared number is 2. While the second participant does know the number with certainty, neither you nor the first participant do, so the output is -1. The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Valentine week has started and Chotu wants to impress his crush. On this rose day Chotu plans to give his crush a bouquet of roses. Chotu visits a rose shop which has N number of roses and each rose has a certain love factor. Now Chotu wants to make a bouquet with maximum number of roses. As Chotu’s crush is very selective, she will only accept the bouquet if it contains roses with distinct love factors. To make the bouquet attractive Chotu wants to arrange roses in the following order. He will first choose rose with the largest love factor followed by the rose with smallest love factor followed by rose with 2^nd largest love factor followed by the rose with 2^nd smallest love factor and so on. Help Chotu to design the bouquet. Input First line contains integer T denoting number of testcases. Second line contains integer N denoting number of roses. Next line contains N integers separated by a single space denoting love factor. Ouptut For each test case print answer in single line in the format explained above. Constraints 1 ≤ T ≤ 10 1 ≤ N ≤ 10^5 1 ≤ Love factor of roses ≤ 10^6 SAMPLE INPUT 2 6 6 2 1 3 2 1 3 1 1 1 SAMPLE OUTPUT 6 1 3 2 1 Explanation In first test case, the distinct roses available are 6 2 1 3. So chotu will make bouquet in following order : 6 1 3 2 In second test case there is only one distinct rose with love factor 1. So he can form a bouquet with only one rose. The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Statement: Security is the major factor, which is prohibiting hackers letting into bank accounts online. But still, we have breached 2nd level security of SBIS bank and a final level is kept for you. Your task is simple, we are receiving an integer token every time we try to send an authorization request. After analysing the integer value, we came to know that after decrypting it, we get a key to SBIS login credential database. How to decrypt: If the integer token contains {2,7,6,9}, print the keys accordingly and skip other digits. If none of the digits from the above set is available, print an empty line. 2 = cde 7 = acf 6 = b3 9 = c6a Input: First line of input contains total number of test cases, T, followed by T lines each containing an integer token, IT. Output: T lines, each containing decrypted value. Constraints: 1 ≤ T ≤ 100 \| 1 ≤ IT ≤ 10^18 SAMPLE INPUT 1 23695632789 SAMPLE OUTPUT cdeb3c6ab3cdeacfc6a Explanation In the sample input, number of test cases is 1 and the corresponding integer token is 23695632789. From the given decrypting method, we get- 2- cde 6- b3 9- c6a and so on. So, the final output is cdeb3c6ab3cdeacfc6a. The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Alice and Bob text each other everyday. Bob, tired of writing long messages has come up with a way to reduce their size. Alice and Bob are both fluent in 2 languages, L1 and L2. A language consists of a collection of distinct words, where each word consists of lowercase letters in the english alphabet. Each word in L1 has a unique translation in L2 and vice versa. To reduce his effort, Bob writes his messages using words from both the languages such that the overall length of the message is minimum. In case the length of a word in both the languages is the same, then Bob writes the word in language L1. Given the content of the message in language L1, find out how Bob will write the message using the above mentioned method. Input: The first line consists of n and m(1 ≤ n ≤ 3000, 1 ≤ m ≤ 3000) - the number of words in the message and the number of words in the language respectively. Each word can be contain at most 10 lowercase letters in the english alphabet. It is guaranteed that no word occurs in both languages and each word occurs in its language exactly once. The following m lines contain the words . The ith line contains 2 strings A[i] and B[i], where A[i] is in L1 and B[i] is in L2. The next line contains n space separated strings from the language L1. Output: n space separated strings that form the message sent by Bob to Alice Sample Input: 5 3 joll wuqrd euzf un hbnyiyc rsoqqveh hbnyiyc joll joll euzf joll Sample Output: hbnyiyc joll joll un joll SAMPLE INPUT 1 1 amit am amit SAMPLE OUTPUT am The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Given is a string S consisting of `0` and `1`. Find the number of strings, modulo 998244353, that can result from applying the following operation on S zero or more times: * Remove the two characters at the beginning of S, erase one of them, and reinsert the other somewhere in S. This operation can be applied only when S has two or more characters. Constraints * 1 \leq \|S\| \leq 300 * S consists of `0` and `1`. Input Input is given from Standard Input in the following format: S Output Print the number of strings, modulo 998244353, that can result from applying the operation on S zero or more times. Examples Input 0001 Output 8 Input 110001 Output 24 Input 11101111011111000000000110000001111100011111000000001111111110000000111111111 Output 697354558 The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. There are N people living on a number line. The i-th person lives at coordinate X_i. You are going to hold a meeting that all N people have to attend. The meeting can be held at any integer coordinate. If you choose to hold the meeting at coordinate P, the i-th person will spend (X_i - P)^2 points of stamina to attend the meeting. Find the minimum total points of stamina the N people have to spend. Constraints * All values in input are integers. * 1 \leq N \leq 100 * 1 \leq X_i \leq 100 Input Input is given from Standard Input in the following format: N X_1 X_2 ... X_N Output Print the minimum total stamina the N people have to spend. Examples Input 2 1 4 Output 5 Input 7 14 14 2 13 56 2 37 Output 2354 The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. We have N ID cards, and there are M gates. We can pass the i-th gate if we have one of the following ID cards: the L_i-th, (L_i+1)-th, ..., and R_i-th ID cards. How many of the ID cards allow us to pass all the gates alone? Constraints * All values in input are integers. * 1 \leq N \leq 10^5 * 1 \leq M \leq 10^5 * 1 \leq L_i \leq R_i \leq N Input Input is given from Standard Input in the following format: N M L_1 R_1 L_2 R_2 \vdots L_M R_M Output Print the number of ID cards that allow us to pass all the gates alone. Examples Input 4 2 1 3 2 4 Output 2 Input 10 3 3 6 5 7 6 9 Output 1 Input 100000 1 1 100000 Output 100000 The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Find the number of integers between 1 and K (inclusive) satisfying the following condition, modulo 10^9 + 7: * The sum of the digits in base ten is a multiple of D. Constraints * All values in input are integers. * 1 \leq K < 10^{10000} * 1 \leq D \leq 100 Input Input is given from Standard Input in the following format: K D Output Print the number of integers satisfying the condition, modulo 10^9 + 7. Examples Input 30 4 Output 6 Input 1000000009 1 Output 2 Input 98765432109876543210 58 Output 635270834 The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. You are given a string S consisting of `0` and `1`. Find the maximum integer K not greater than \|S\| such that we can turn all the characters of S into `0` by repeating the following operation some number of times. * Choose a contiguous segment [l,r] in S whose length is at least K (that is, r-l+1\geq K must be satisfied). For each integer i such that l\leq i\leq r, do the following: if S_i is `0`, replace it with `1`; if S_i is `1`, replace it with `0`. Constraints * 1\leq \|S\|\leq 10^5 * S_i(1\leq i\leq N) is either `0` or `1`. Input Input is given from Standard Input in the following format: S Output Print the maximum integer K such that we can turn all the characters of S into `0` by repeating the operation some number of times. Examples Input 010 Output 2 Input 100000000 Output 8 Input 00001111 Output 4 The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.

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