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Solve the programming task below in a Python markdown code block. You are playing a computer card game called Splay the Sire. Currently you are struggling to defeat the final boss of the game. The boss battle consists of n turns. During each turn, you will get several cards. Each card has two parameters: its cost c_i and damage d_i. You may play some of your cards during each turn in some sequence (you choose the cards and the exact order they are played), as long as the total cost of the cards you play during the turn does not exceed 3. After playing some (possibly zero) cards, you end your turn, and all cards you didn't play are discarded. Note that you can use each card at most once. Your character has also found an artifact that boosts the damage of some of your actions: every 10-th card you play deals double damage. What is the maximum possible damage you can deal during n turns? Input The first line contains one integer n (1 ≤ n ≤ 2 ⋅ 10^5) — the number of turns. Then n blocks of input follow, the i-th block representing the cards you get during the i-th turn. Each block begins with a line containing one integer k_i (1 ≤ k_i ≤ 2 ⋅ 10^5) — the number of cards you get during i-th turn. Then k_i lines follow, each containing two integers c_j and d_j (1 ≤ c_j ≤ 3, 1 ≤ d_j ≤ 10^9) — the parameters of the corresponding card. It is guaranteed that ∑ _{i = 1}^{n} k_i ≤ 2 ⋅ 10^5. Output Print one integer — the maximum damage you may deal. Example Input 5 3 1 6 1 7 1 5 2 1 4 1 3 3 1 10 3 5 2 3 3 1 15 2 4 1 10 1 1 100 Output 263 Note In the example test the best course of action is as follows: During the first turn, play all three cards in any order and deal 18 damage. During the second turn, play both cards and deal 7 damage. During the third turn, play the first and the third card and deal 13 damage. During the fourth turn, play the first and the third card and deal 25 damage. During the fifth turn, play the only card, which will deal double damage (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. This problem differs from the previous one only in the absence of the constraint on the equal length of all numbers a_1, a_2, ..., a_n. A team of SIS students is going to make a trip on a submarine. Their target is an ancient treasure in a sunken ship lying on the bottom of the Great Rybinsk sea. Unfortunately, the students don't know the coordinates of the ship, so they asked Meshanya (who is a hereditary mage) to help them. He agreed to help them, but only if they solve his problem. Let's denote a function that alternates digits of two numbers f(a_1 a_2 ... a_{p - 1} a_p, b_1 b_2 ... b_{q - 1} b_q), where a_1 ... a_p and b_1 ... b_q are digits of two integers written in the decimal notation without leading zeros. In other words, the function f(x, y) alternately shuffles the digits of the numbers x and y by writing them from the lowest digits to the older ones, starting with the number y. The result of the function is also built from right to left (that is, from the lower digits to the older ones). If the digits of one of the arguments have ended, then the remaining digits of the other argument are written out. Familiarize with examples and formal definitions of the function below. For example: $$$f(1111, 2222) = 12121212 f(7777, 888) = 7787878 f(33, 44444) = 4443434 f(555, 6) = 5556 f(111, 2222) = 2121212$$$ Formally, * if p ≥ q then f(a_1 ... a_p, b_1 ... b_q) = a_1 a_2 ... a_{p - q + 1} b_1 a_{p - q + 2} b_2 ... a_{p - 1} b_{q - 1} a_p b_q; * if p < q then f(a_1 ... a_p, b_1 ... b_q) = b_1 b_2 ... b_{q - p} a_1 b_{q - p + 1} a_2 ... a_{p - 1} b_{q - 1} a_p b_q. Mishanya gives you an array consisting of n integers a_i, your task is to help students to calculate ∑_{i = 1}^{n}∑_{j = 1}^{n} f(a_i, a_j) modulo 998 244 353. Input The first line of the input contains a single integer n (1 ≤ n ≤ 100 000) — the number of elements in the array. The second line of the input contains n integers a_1, a_2, ..., a_n (1 ≤ a_i ≤ 10^9) — the elements of the array. Output Print the answer modulo 998 244 353. Examples Input 3 12 3 45 Output 12330 Input 2 123 456 Output 1115598 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 is playing with some stones. Now there are three numbered heaps of stones. The first of them contains a stones, the second of them contains b stones and the third of them contains c stones. Each time she can do one of two operations: 1. take one stone from the first heap and two stones from the second heap (this operation can be done only if the first heap contains at least one stone and the second heap contains at least two stones); 2. take one stone from the second heap and two stones from the third heap (this operation can be done only if the second heap contains at least one stone and the third heap contains at least two stones). She wants to get the maximum number of stones, but she doesn't know what to do. Initially, she has 0 stones. Can you help her? Input The first line contains one integer t (1 ≤ t ≤ 100) — the number of test cases. Next t lines describe test cases in the following format: Line contains three non-negative integers a, b and c, separated by spaces (0 ≤ a,b,c ≤ 100) — the number of stones in the first, the second and the third heap, respectively. In hacks it is allowed to use only one test case in the input, so t = 1 should be satisfied. Output Print t lines, the answers to the test cases in the same order as in the input. The answer to the test case is the integer — the maximum possible number of stones that Alice can take after making some operations. Example Input 3 3 4 5 1 0 5 5 3 2 Output 9 0 6 Note For the first test case in the first test, Alice can take two stones from the second heap and four stones from the third heap, making the second operation two times. Then she can take one stone from the first heap and two stones from the second heap, making the first operation one time. The summary number of stones, that Alice will take is 9. It is impossible to make some operations to take more than 9 stones, so the answer is 9. 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. Piet is one of the most known visual esoteric programming languages. The programs in Piet are constructed from colorful blocks of pixels and interpreted using pretty complicated rules. In this problem we will use a subset of Piet language with simplified rules. The program will be a rectangular image consisting of colored and black pixels. The color of each pixel will be given by an integer number between 0 and 9, inclusive, with 0 denoting black. A block of pixels is defined as a rectangle of pixels of the same color (not black). It is guaranteed that all connected groups of colored pixels of the same color will form rectangular blocks. Groups of black pixels can form arbitrary shapes. The program is interpreted using movement of instruction pointer (IP) which consists of three parts: * current block pointer (BP); note that there is no concept of current pixel within the block; * direction pointer (DP) which can point left, right, up or down; * block chooser (CP) which can point to the left or to the right from the direction given by DP; in absolute values CP can differ from DP by 90 degrees counterclockwise or clockwise, respectively. Initially BP points to the block which contains the top-left corner of the program, DP points to the right, and CP points to the left (see the orange square on the image below). One step of program interpretation changes the state of IP in a following way. The interpreter finds the furthest edge of the current color block in the direction of the DP. From all pixels that form this edge, the interpreter selects the furthest one in the direction of CP. After this, BP attempts to move from this pixel into the next one in the direction of DP. If the next pixel belongs to a colored block, this block becomes the current one, and two other parts of IP stay the same. It the next pixel is black or outside of the program, BP stays the same but two other parts of IP change. If CP was pointing to the left, now it points to the right, and DP stays the same. If CP was pointing to the right, now it points to the left, and DP is rotated 90 degrees clockwise. This way BP will never point to a black block (it is guaranteed that top-left pixel of the program will not be black). You are given a Piet program. You have to figure out which block of the program will be current after n steps. Input The first line of the input contains two integer numbers m (1 ≤ m ≤ 50) and n (1 ≤ n ≤ 5·107). Next m lines contain the rows of the program. All the lines have the same length between 1 and 50 pixels, and consist of characters 0-9. The first character of the first line will not be equal to 0. Output Output the color of the block which will be current after n steps of program interpretation. Examples Input 2 10 12 43 Output 1 Input 3 12 1423 6624 6625 Output 6 Input 5 9 10345 23456 34567 45678 56789 Output 5 Note In the first example IP changes in the following way. After step 1 block 2 becomes current one and stays it after two more steps. After step 4 BP moves to block 3, after step 7 — to block 4, and finally after step 10 BP returns to block 1. <image> The sequence of states of IP is shown on the image: the arrows are traversed clockwise, the main arrow shows direction of DP, the side one — the direction of CP. 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 arrays of integers a_1,…,a_n and b_1,…,b_m. Your task is to find a non-empty array c_1,…,c_k that is a subsequence of a_1,…,a_n, and also a subsequence of b_1,…,b_m. If there are multiple answers, find one of the smallest possible length. If there are still multiple of the smallest possible length, find any. If there are no such arrays, you should report about it. A sequence a is a subsequence of a sequence b if a can be obtained from b by deletion of several (possibly, zero) elements. For example, [3,1] is a subsequence of [3,2,1] and [4,3,1], but not a subsequence of [1,3,3,7] and [3,10,4]. Input The first line contains a single integer t (1≤ t≤ 1000) — the number of test cases. Next 3t lines contain descriptions of test cases. The first line of each test case contains two integers n and m (1≤ n,m≤ 1000) — the lengths of the two arrays. The second line of each test case contains n integers a_1,…,a_n (1≤ a_i≤ 1000) — the elements of the first array. The third line of each test case contains m integers b_1,…,b_m (1≤ b_i≤ 1000) — the elements of the second array. It is guaranteed that the sum of n and the sum of m across all test cases does not exceed 1000 (∑_{i=1}^t n_i, ∑_{i=1}^t m_i≤ 1000). Output For each test case, output "YES" if a solution exists, or "NO" otherwise. If the answer is "YES", on the next line output an integer k (1≤ k≤ 1000) — the length of the array, followed by k integers c_1,…,c_k (1≤ c_i≤ 1000) — the elements of the array. If there are multiple solutions with the smallest possible k, output any. Example Input 5 4 5 10 8 6 4 1 2 3 4 5 1 1 3 3 1 1 3 2 5 3 1000 2 2 2 3 3 1 5 5 5 1 2 3 4 5 1 2 3 4 5 Output YES 1 4 YES 1 3 NO YES 1 3 YES 1 2 Note In the first test case, [4] is a subsequence of [10, 8, 6, 4] and [1, 2, 3, 4, 5]. This array has length 1, it is the smallest possible length of a subsequence of both a and b. In the third test case, no non-empty subsequences of both [3] and [2] exist, so the answer is "NO". 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. Valeric and Valerko missed the last Euro football game, so they decided to watch the game's key moments on the Net. They want to start watching as soon as possible but the connection speed is too low. If they turn on the video right now, it will "hang up" as the size of data to watch per second will be more than the size of downloaded data per second. The guys want to watch the whole video without any pauses, so they have to wait some integer number of seconds for a part of the video to download. After this number of seconds passes, they can start watching. Waiting for the whole video to download isn't necessary as the video can download after the guys started to watch. Let's suppose that video's length is c seconds and Valeric and Valerko wait t seconds before the watching. Then for any moment of time t0, t ≤ t0 ≤ c + t, the following condition must fulfill: the size of data received in t0 seconds is not less than the size of data needed to watch t0 - t seconds of the video. Of course, the guys want to wait as little as possible, so your task is to find the minimum integer number of seconds to wait before turning the video on. The guys must watch the video without pauses. Input The first line contains three space-separated integers a, b and c (1 ≤ a, b, c ≤ 1000, a > b). The first number (a) denotes the size of data needed to watch one second of the video. The second number (b) denotes the size of data Valeric and Valerko can download from the Net per second. The third number (c) denotes the video's length in seconds. Output Print a single number — the minimum integer number of seconds that Valeric and Valerko must wait to watch football without pauses. Examples Input 4 1 1 Output 3 Input 10 3 2 Output 5 Input 13 12 1 Output 1 Note In the first sample video's length is 1 second and it is necessary 4 units of data for watching 1 second of video, so guys should download 4 · 1 = 4 units of data to watch the whole video. The most optimal way is to wait 3 seconds till 3 units of data will be downloaded and then start watching. While guys will be watching video 1 second, one unit of data will be downloaded and Valerik and Valerko will have 4 units of data by the end of watching. Also every moment till the end of video guys will have more data then necessary for watching. In the second sample guys need 2 · 10 = 20 units of data, so they have to wait 5 seconds and after that they will have 20 units before the second second ends. However, if guys wait 4 seconds, they will be able to watch first second of video without pauses, but they will download 18 units of data by the end of second second and it is less then necessary. 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 black king is standing on a chess field consisting of 109 rows and 109 columns. We will consider the rows of the field numbered with integers from 1 to 109 from top to bottom. The columns are similarly numbered with integers from 1 to 109 from left to right. We will denote a cell of the field that is located in the i-th row and j-th column as (i, j). You know that some squares of the given chess field are allowed. All allowed cells of the chess field are given as n segments. Each segment is described by three integers ri, ai, bi (ai ≤ bi), denoting that cells in columns from number ai to number bi inclusive in the ri-th row are allowed. Your task is to find the minimum number of moves the king needs to get from square (x0, y0) to square (x1, y1), provided that he only moves along the allowed cells. In other words, the king can be located only on allowed cells on his way. Let us remind you that a chess king can move to any of the neighboring cells in one move. Two cells of a chess field are considered neighboring if they share at least one point. Input The first line contains four space-separated integers x0, y0, x1, y1 (1 ≤ x0, y0, x1, y1 ≤ 109), denoting the initial and the final positions of the king. The second line contains a single integer n (1 ≤ n ≤ 105), denoting the number of segments of allowed cells. Next n lines contain the descriptions of these segments. The i-th line contains three space-separated integers ri, ai, bi (1 ≤ ri, ai, bi ≤ 109, ai ≤ bi), denoting that cells in columns from number ai to number bi inclusive in the ri-th row are allowed. Note that the segments of the allowed cells can intersect and embed arbitrarily. It is guaranteed that the king's initial and final position are allowed cells. It is guaranteed that the king's initial and the final positions do not coincide. It is guaranteed that the total length of all given segments doesn't exceed 105. Output If there is no path between the initial and final position along allowed cells, print -1. Otherwise print a single integer — the minimum number of moves the king needs to get from the initial position to the final one. Examples Input 5 7 6 11 3 5 3 8 6 7 11 5 2 5 Output 4 Input 3 4 3 10 3 3 1 4 4 5 9 3 10 10 Output 6 Input 1 1 2 10 2 1 1 3 2 6 10 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. Manao works on a sports TV. He's spent much time watching the football games of some country. After a while he began to notice different patterns. For example, each team has two sets of uniforms: home uniform and guest uniform. When a team plays a game at home, the players put on the home uniform. When a team plays as a guest on somebody else's stadium, the players put on the guest uniform. The only exception to that rule is: when the home uniform color of the host team matches the guests' uniform, the host team puts on its guest uniform as well. For each team the color of the home and guest uniform is different. There are n teams taking part in the national championship. The championship consists of n·(n - 1) games: each team invites each other team to its stadium. At this point Manao wondered: how many times during the championship is a host team going to put on the guest uniform? Note that the order of the games does not affect this number. You know the colors of the home and guest uniform for each team. For simplicity, the colors are numbered by integers in such a way that no two distinct colors have the same number. Help Manao find the answer to his question. Input The first line contains an integer n (2 ≤ n ≤ 30). Each of the following n lines contains a pair of distinct space-separated integers hi, ai (1 ≤ hi, ai ≤ 100) — the colors of the i-th team's home and guest uniforms, respectively. Output In a single line print the number of games where the host team is going to play in the guest uniform. Examples Input 3 1 2 2 4 3 4 Output 1 Input 4 100 42 42 100 5 42 100 5 Output 5 Input 2 1 2 1 2 Output 0 Note In the first test case the championship consists of 6 games. The only game with the event in question is the game between teams 2 and 1 on the stadium of team 2. In the second test sample the host team will have to wear guest uniform in the games between teams: 1 and 2, 2 and 1, 2 and 3, 3 and 4, 4 and 2 (the host team is written first). 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> Input The first line of the input is a string (between 1 and 50 characters long, inclusive). Each character will be a letter of English alphabet, lowercase or uppercase. The second line of the input is an integer between 0 and 26, inclusive. Output Output the required string. Examples Input AprilFool 14 Output AprILFooL 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 the rush of modern life, people often forget how beautiful the world is. The time to enjoy those around them is so little that some even stand in queues to several rooms at the same time in the clinic, running from one queue to another. (Cultural note: standing in huge and disorganized queues for hours is a native tradition in Russia, dating back to the Soviet period. Queues can resemble crowds rather than lines. Not to get lost in such a queue, a person should follow a strict survival technique: you approach the queue and ask who the last person is, somebody answers and you join the crowd. Now you're the last person in the queue till somebody else shows up. You keep an eye on the one who was last before you as he is your only chance to get to your destination) I'm sure many people have had the problem when a stranger asks who the last person in the queue is and even dares to hint that he will be the last in the queue and then bolts away to some unknown destination. These are the representatives of the modern world, in which the ratio of lack of time is so great that they do not even watch foreign top-rated TV series. Such people often create problems in queues, because the newcomer does not see the last person in the queue and takes a place after the "virtual" link in this chain, wondering where this legendary figure has left. The Smart Beaver has been ill and he's made an appointment with a therapist. The doctor told the Beaver the sad news in a nutshell: it is necessary to do an electrocardiogram. The next day the Smart Beaver got up early, put on the famous TV series on download (three hours till the download's complete), clenched his teeth and bravely went to join a queue to the electrocardiogram room, which is notorious for the biggest queues at the clinic. Having stood for about three hours in the queue, the Smart Beaver realized that many beavers had not seen who was supposed to stand in the queue before them and there was a huge mess. He came up to each beaver in the ECG room queue and asked who should be in front of him in the queue. If the beaver did not know his correct position in the queue, then it might be his turn to go get an ECG, or maybe he should wait for a long, long time... As you've guessed, the Smart Beaver was in a hurry home, so he gave you all the necessary information for you to help him to determine what his number in the queue can be. Input The first line contains two integers n (1 ≤ n ≤ 103) and x (1 ≤ x ≤ n) — the number of beavers that stand in the queue and the Smart Beaver's number, correspondingly. All willing to get to the doctor are numbered from 1 to n. The second line contains n integers a1, a2, ..., an (0 ≤ ai ≤ n) — the number of the beaver followed by the i-th beaver. If ai = 0, then the i-th beaver doesn't know who is should be in front of him. It is guaranteed that values ai are correct. That is there is no cycles in the dependencies. And any beaver is followed by at most one beaver in the queue. The input limits for scoring 30 points are (subproblem B1): * It is guaranteed that the number of zero elements ai doesn't exceed 20. The input limits for scoring 100 points are (subproblems B1+B2): * The number of zero elements ai is arbitrary. Output Print all possible positions of the Smart Beaver in the line in the increasing order. Examples Input 6 1 2 0 4 0 6 0 Output 2 4 6 Input 6 2 2 3 0 5 6 0 Output 2 5 Input 4 1 0 0 0 0 Output 1 2 3 4 Input 6 2 0 0 1 0 4 5 Output 1 3 4 6 Note <image> Picture for the fourth 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. Xenia the beginner mathematician is a third year student at elementary school. She is now learning the addition operation. The teacher has written down the sum of multiple numbers. Pupils should calculate the sum. To make the calculation easier, the sum only contains numbers 1, 2 and 3. Still, that isn't enough for Xenia. She is only beginning to count, so she can calculate a sum only if the summands follow in non-decreasing order. For example, she can't calculate sum 1+3+2+1 but she can calculate sums 1+1+2 and 3+3. You've got the sum that was written on the board. Rearrange the summans and print the sum in such a way that Xenia can calculate the sum. Input The first line contains a non-empty string s — the sum Xenia needs to count. String s contains no spaces. It only contains digits and characters "+". Besides, string s is a correct sum of numbers 1, 2 and 3. String s is at most 100 characters long. Output Print the new sum that Xenia can count. Examples Input 3+2+1 Output 1+2+3 Input 1+1+3+1+3 Output 1+1+1+3+3 Input 2 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. Levko loves array a1, a2, ... , an, consisting of integers, very much. That is why Levko is playing with array a, performing all sorts of operations with it. Each operation Levko performs is of one of two types: 1. Increase all elements from li to ri by di. In other words, perform assignments aj = aj + di for all j that meet the inequation li ≤ j ≤ ri. 2. Find the maximum of elements from li to ri. That is, calculate the value <image>. Sadly, Levko has recently lost his array. Fortunately, Levko has records of all operations he has performed on array a. Help Levko, given the operation records, find at least one suitable array. The results of all operations for the given array must coincide with the record results. Levko clearly remembers that all numbers in his array didn't exceed 109 in their absolute value, so he asks you to find such an array. Input The first line contains two integers n and m (1 ≤ n, m ≤ 5000) — the size of the array and the number of operations in Levko's records, correspondingly. Next m lines describe the operations, the i-th line describes the i-th operation. The first integer in the i-th line is integer ti (1 ≤ ti ≤ 2) that describes the operation type. If ti = 1, then it is followed by three integers li, ri and di (1 ≤ li ≤ ri ≤ n, - 104 ≤ di ≤ 104) — the description of the operation of the first type. If ti = 2, then it is followed by three integers li, ri and mi (1 ≤ li ≤ ri ≤ n, - 5·107 ≤ mi ≤ 5·107) — the description of the operation of the second type. The operations are given in the order Levko performed them on his array. Output In the first line print "YES" (without the quotes), if the solution exists and "NO" (without the quotes) otherwise. If the solution exists, then on the second line print n integers a1, a2, ... , an (\|ai\| ≤ 109) — the recovered array. Examples Input 4 5 1 2 3 1 2 1 2 8 2 3 4 7 1 1 3 3 2 3 4 8 Output YES 4 7 4 7 Input 4 5 1 2 3 1 2 1 2 8 2 3 4 7 1 1 3 3 2 3 4 13 Output NO 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 bear decided to store some raspberry for the winter. He cunningly found out the price for a barrel of honey in kilos of raspberry for each of the following n days. According to the bear's data, on the i-th (1 ≤ i ≤ n) day, the price for one barrel of honey is going to is xi kilos of raspberry. Unfortunately, the bear has neither a honey barrel, nor the raspberry. At the same time, the bear's got a friend who is ready to lend him a barrel of honey for exactly one day for c kilograms of raspberry. That's why the bear came up with a smart plan. He wants to choose some day d (1 ≤ d < n), lent a barrel of honey and immediately (on day d) sell it according to a daily exchange rate. The next day (d + 1) the bear wants to buy a new barrel of honey according to a daily exchange rate (as he's got some raspberry left from selling the previous barrel) and immediately (on day d + 1) give his friend the borrowed barrel of honey as well as c kilograms of raspberry for renting the barrel. The bear wants to execute his plan at most once and then hibernate. What maximum number of kilograms of raspberry can he earn? Note that if at some point of the plan the bear runs out of the raspberry, then he won't execute such a plan. Input The first line contains two space-separated integers, n and c (2 ≤ n ≤ 100, 0 ≤ c ≤ 100), — the number of days and the number of kilos of raspberry that the bear should give for borrowing the barrel. The second line contains n space-separated integers x1, x2, ..., xn (0 ≤ xi ≤ 100), the price of a honey barrel on day i. Output Print a single integer — the answer to the problem. Examples Input 5 1 5 10 7 3 20 Output 3 Input 6 2 100 1 10 40 10 40 Output 97 Input 3 0 1 2 3 Output 0 Note In the first sample the bear will lend a honey barrel at day 3 and then sell it for 7. Then the bear will buy a barrel for 3 and return it to the friend. So, the profit is (7 - 3 - 1) = 3. In the second sample bear will lend a honey barrel at day 1 and then sell it for 100. Then the bear buy the barrel for 1 at the day 2. So, the profit is (100 - 1 - 2) = 97. 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. Little Chris is participating in a graph cutting contest. He's a pro. The time has come to test his skills to the fullest. Chris is given a simple undirected connected graph with n vertices (numbered from 1 to n) and m edges. The problem is to cut it into edge-distinct paths of length 2. Formally, Chris has to partition all edges of the graph into pairs in such a way that the edges in a single pair are adjacent and each edge must be contained in exactly one pair. For example, the figure shows a way Chris can cut a graph. The first sample test contains the description of this graph. <image> You are given a chance to compete with Chris. Find a way to cut the given graph or determine that it is impossible! Input The first line of input contains two space-separated integers n and m (1 ≤ n, m ≤ 105), the number of vertices and the number of edges in the graph. The next m lines contain the description of the graph's edges. The i-th line contains two space-separated integers ai and bi (1 ≤ ai, bi ≤ n; ai ≠ bi), the numbers of the vertices connected by the i-th edge. It is guaranteed that the given graph is simple (without self-loops and multi-edges) and connected. Note: since the size of the input and output could be very large, don't use slow output techniques in your language. For example, do not use input and output streams (cin, cout) in C++. Output If it is possible to cut the given graph into edge-distinct paths of length 2, output <image> lines. In the i-th line print three space-separated integers xi, yi and zi, the description of the i-th path. The graph should contain this path, i.e., the graph should contain edges (xi, yi) and (yi, zi). Each edge should appear in exactly one path of length 2. If there are multiple solutions, output any of them. If it is impossible to cut the given graph, print "No solution" (without quotes). Examples Input 8 12 1 2 2 3 3 4 4 1 1 3 2 4 3 5 3 6 5 6 6 7 6 8 7 8 Output 1 2 4 1 3 2 1 4 3 5 3 6 5 6 8 6 7 8 Input 3 3 1 2 2 3 3 1 Output No solution Input 3 2 1 2 2 3 Output 1 2 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. There are five people playing a game called "Generosity". Each person gives some non-zero number of coins b as an initial bet. After all players make their bets of b coins, the following operation is repeated for several times: a coin is passed from one player to some other player. Your task is to write a program that can, given the number of coins each player has at the end of the game, determine the size b of the initial bet or find out that such outcome of the game cannot be obtained for any positive number of coins b in the initial bet. Input The input consists of a single line containing five integers c1, c2, c3, c4 and c5 — the number of coins that the first, second, third, fourth and fifth players respectively have at the end of the game (0 ≤ c1, c2, c3, c4, c5 ≤ 100). Output Print the only line containing a single positive integer b — the number of coins in the initial bet of each player. If there is no such value of b, then print the only value "-1" (quotes for clarity). Examples Input 2 5 4 0 4 Output 3 Input 4 5 9 2 1 Output -1 Note In the first sample the following sequence of operations is possible: 1. One coin is passed from the fourth player to the second player; 2. One coin is passed from the fourth player to the fifth player; 3. One coin is passed from the first player to the third player; 4. One coin is passed from the fourth player to the second player. 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. New Year is coming, and Jaehyun decided to read many books during 2015, unlike this year. He has n books numbered by integers from 1 to n. The weight of the i-th (1 ≤ i ≤ n) book is wi. As Jaehyun's house is not large enough to have a bookshelf, he keeps the n books by stacking them vertically. When he wants to read a certain book x, he follows the steps described below. 1. He lifts all the books above book x. 2. He pushes book x out of the stack. 3. He puts down the lifted books without changing their order. 4. After reading book x, he puts book x on the top of the stack. <image> He decided to read books for m days. In the j-th (1 ≤ j ≤ m) day, he will read the book that is numbered with integer bj (1 ≤ bj ≤ n). To read the book, he has to use the process described in the paragraph above. It is possible that he decides to re-read the same book several times. After making this plan, he realized that the total weight of books he should lift during m days would be too heavy. So, he decided to change the order of the stacked books before the New Year comes, and minimize the total weight. You may assume that books can be stacked in any possible order. Note that book that he is going to read on certain step isn't considered as lifted on that step. Can you help him? Input The first line contains two space-separated integers n (2 ≤ n ≤ 500) and m (1 ≤ m ≤ 1000) — the number of books, and the number of days for which Jaehyun would read books. The second line contains n space-separated integers w1, w2, ..., wn (1 ≤ wi ≤ 100) — the weight of each book. The third line contains m space separated integers b1, b2, ..., bm (1 ≤ bj ≤ n) — the order of books that he would read. Note that he can read the same book more than once. Output Print the minimum total weight of books he should lift, which can be achieved by rearranging the order of stacked books. Examples Input 3 5 1 2 3 1 3 2 3 1 Output 12 Note Here's a picture depicting the example. Each vertical column presents the stacked books. <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. In this problem you will meet the simplified model of game King of Thieves. In a new ZeptoLab game called "King of Thieves" your aim is to reach a chest with gold by controlling your character, avoiding traps and obstacles on your way. <image> An interesting feature of the game is that you can design your own levels that will be available to other players. Let's consider the following simple design of a level. A dungeon consists of n segments located at a same vertical level, each segment is either a platform that character can stand on, or a pit with a trap that makes player lose if he falls into it. All segments have the same length, platforms on the scheme of the level are represented as '' and pits are represented as '.'. One of things that affects speedrun characteristics of the level is a possibility to perform a series of consecutive jumps of the same length. More formally, when the character is on the platform number i1, he can make a sequence of jumps through the platforms i1 < i2 < ... < ik, if i2 - i1 = i3 - i2 = ... = ik - ik - 1. Of course, all segments i1, i2, ... ik should be exactly the platforms, not pits. Let's call a level to be good if you can perform a sequence of four jumps of the same length or in the other words there must be a sequence i1, i2, ..., i5, consisting of five platforms so that the intervals between consecutive platforms are of the same length. Given the scheme of the level, check if it is good. Input The first line contains integer n (1 ≤ n ≤ 100) — the number of segments on the level. Next line contains the scheme of the level represented as a string of n characters '' and '.'. Output If the level is good, print the word "yes" (without the quotes), otherwise print the word "no" (without the quotes). Examples Input 16 .*...... Output yes Input 11 ....... Output no Note In the first sample test you may perform a sequence of jumps through platforms 2, 5, 8, 11, 14. 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 of n integers a1, a2, ..., an. Determine a real number x such that the weakness of the sequence a1 - x, a2 - x, ..., an - x is as small as possible. The weakness of a sequence is defined as the maximum value of the poorness over all segments (contiguous subsequences) of a sequence. The poorness of a segment is defined as the absolute value of sum of the elements of segment. Input The first line contains one integer n (1 ≤ n ≤ 200 000), the length of a sequence. The second line contains n integers a1, a2, ..., an (\|ai\| ≤ 10 000). Output Output a real number denoting the minimum possible weakness of a1 - x, a2 - x, ..., an - x. Your answer will be considered correct if its relative or absolute error doesn't exceed 10 - 6. Examples Input 3 1 2 3 Output 1.000000000000000 Input 4 1 2 3 4 Output 2.000000000000000 Input 10 1 10 2 9 3 8 4 7 5 6 Output 4.500000000000000 Note For the first case, the optimal value of x is 2 so the sequence becomes - 1, 0, 1 and the max poorness occurs at the segment "-1" or segment "1". The poorness value (answer) equals to 1 in this case. For the second sample the optimal value of x is 2.5 so the sequence becomes - 1.5, - 0.5, 0.5, 1.5 and the max poorness occurs on segment "-1.5 -0.5" or "0.5 1.5". The poorness value (answer) equals to 2 in this case. 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. Polycarp is working on a new project called "Polychat". Following modern tendencies in IT, he decided, that this project should contain chat as well. To achieve this goal, Polycarp has spent several hours in front of his laptop and implemented a chat server that can process three types of commands: * Include a person to the chat ('Add' command). * Remove a person from the chat ('Remove' command). * Send a message from a person to all people, who are currently in the chat, including the one, who sends the message ('Send' command). Now Polycarp wants to find out the amount of outgoing traffic that the server will produce while processing a particular set of commands. Polycarp knows that chat server sends no traffic for 'Add' and 'Remove' commands. When 'Send' command is processed, server sends l bytes to each participant of the chat, where l is the length of the message. As Polycarp has no time, he is asking for your help in solving this problem. Input Input file will contain not more than 100 commands, each in its own line. No line will exceed 100 characters. Formats of the commands will be the following: * +<name> for 'Add' command. * -<name> for 'Remove' command. * <sender_name>:<message_text> for 'Send' command. <name> and <sender_name> is a non-empty sequence of Latin letters and digits. <message_text> can contain letters, digits and spaces, but can't start or end with a space. <message_text> can be an empty line. It is guaranteed, that input data are correct, i.e. there will be no 'Add' command if person with such a name is already in the chat, there will be no 'Remove' command if there is no person with such a name in the chat etc. All names are case-sensitive. Output Print a single number — answer to the problem. Examples Input +Mike Mike:hello +Kate +Dmitry -Dmitry Kate:hi -Kate Output 9 Input +Mike -Mike +Mike Mike:Hi I am here -Mike +Kate -Kate Output 14 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. Wet Shark asked Rat Kwesh to generate three positive real numbers x, y and z, from 0.1 to 200.0, inclusive. Wet Krash wants to impress Wet Shark, so all generated numbers will have exactly one digit after the decimal point. Wet Shark knows Rat Kwesh will want a lot of cheese. So he will give the Rat an opportunity to earn a lot of cheese. He will hand the three numbers x, y and z to Rat Kwesh, and Rat Kwesh will pick one of the these twelve options: 1. a1 = xyz; 2. a2 = xzy; 3. a3 = (xy)z; 4. a4 = (xz)y; 5. a5 = yxz; 6. a6 = yzx; 7. a7 = (yx)z; 8. a8 = (yz)x; 9. a9 = zxy; 10. a10 = zyx; 11. a11 = (zx)y; 12. a12 = (zy)x. Let m be the maximum of all the ai, and c be the smallest index (from 1 to 12) such that ac = m. Rat's goal is to find that c, and he asks you to help him. Rat Kwesh wants to see how much cheese he gets, so he you will have to print the expression corresponding to that ac. Input The only line of the input contains three space-separated real numbers x, y and z (0.1 ≤ x, y, z ≤ 200.0). Each of x, y and z is given with exactly one digit after the decimal point. Output Find the maximum value of expression among xyz, xzy, (xy)z, (xz)y, yxz, yzx, (yx)z, (yz)x, zxy, zyx, (zx)y, (zy)x and print the corresponding expression. If there are many maximums, print the one that comes first in the list. xyz should be outputted as x^y^z (without brackets), and (xy)z should be outputted as (x^y)^z (quotes for clarity). Examples Input 1.1 3.4 2.5 Output z^y^x Input 2.0 2.0 2.0 Output x^y^z Input 1.9 1.8 1.7 Output (x^y)^z 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. Bearland has n cities, numbered 1 through n. Cities are connected via bidirectional roads. Each road connects two distinct cities. No two roads connect the same pair of cities. Bear Limak was once in a city a and he wanted to go to a city b. There was no direct connection so he decided to take a long walk, visiting each city exactly once. Formally: * There is no road between a and b. * There exists a sequence (path) of n distinct cities v1, v2, ..., vn that v1 = a, vn = b and there is a road between vi and vi + 1 for <image>. On the other day, the similar thing happened. Limak wanted to travel between a city c and a city d. There is no road between them but there exists a sequence of n distinct cities u1, u2, ..., un that u1 = c, un = d and there is a road between ui and ui + 1 for <image>. Also, Limak thinks that there are at most k roads in Bearland. He wonders whether he remembers everything correctly. Given n, k and four distinct cities a, b, c, d, can you find possible paths (v1, ..., vn) and (u1, ..., un) to satisfy all the given conditions? Find any solution or print -1 if it's impossible. Input The first line of the input contains two integers n and k (4 ≤ n ≤ 1000, n - 1 ≤ k ≤ 2n - 2) — the number of cities and the maximum allowed number of roads, respectively. The second line contains four distinct integers a, b, c and d (1 ≤ a, b, c, d ≤ n). Output Print -1 if it's impossible to satisfy all the given conditions. Otherwise, print two lines with paths descriptions. The first of these two lines should contain n distinct integers v1, v2, ..., vn where v1 = a and vn = b. The second line should contain n distinct integers u1, u2, ..., un where u1 = c and un = d. Two paths generate at most 2n - 2 roads: (v1, v2), (v2, v3), ..., (vn - 1, vn), (u1, u2), (u2, u3), ..., (un - 1, un). Your answer will be considered wrong if contains more than k distinct roads or any other condition breaks. Note that (x, y) and (y, x) are the same road. Examples Input 7 11 2 4 7 3 Output 2 7 1 3 6 5 4 7 1 5 4 6 2 3 Input 1000 999 10 20 30 40 Output -1 Note In the first sample test, there should be 7 cities and at most 11 roads. The provided sample solution generates 10 roads, as in the drawing. You can also see a simple path of length n between 2 and 4, and a path between 7 and 3. <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. In late autumn evening n robots gathered in the cheerful company of friends. Each robot has a unique identifier — an integer from 1 to 109. At some moment, robots decided to play the game "Snowball". Below there are the rules of this game. First, all robots stand in a row. Then the first robot says his identifier. After that the second robot says the identifier of the first robot and then says his own identifier. Then the third robot says the identifier of the first robot, then says the identifier of the second robot and after that says his own. This process continues from left to right until the n-th robot says his identifier. Your task is to determine the k-th identifier to be pronounced. Input The first line contains two positive integers n and k (1 ≤ n ≤ 100 000, 1 ≤ k ≤ min(2·109, n·(n + 1) / 2). The second line contains the sequence id1, id2, ..., idn (1 ≤ idi ≤ 109) — identifiers of roborts. It is guaranteed that all identifiers are different. Output Print the k-th pronounced identifier (assume that the numeration starts from 1). Examples Input 2 2 1 2 Output 1 Input 4 5 10 4 18 3 Output 4 Note In the first sample identifiers of robots will be pronounced in the following order: 1, 1, 2. As k = 2, the answer equals to 1. In the second test case identifiers of robots will be pronounced in the following order: 10, 10, 4, 10, 4, 18, 10, 4, 18, 3. As k = 5, the answer equals to 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. You are given a permutation of the numbers 1, 2, ..., n and m pairs of positions (aj, bj). At each step you can choose a pair from the given positions and swap the numbers in that positions. What is the lexicographically maximal permutation one can get? Let p and q be two permutations of the numbers 1, 2, ..., n. p is lexicographically smaller than the q if a number 1 ≤ i ≤ n exists, so pk = qk for 1 ≤ k < i and pi < qi. Input The first line contains two integers n and m (1 ≤ n, m ≤ 106) — the length of the permutation p and the number of pairs of positions. The second line contains n distinct integers pi (1 ≤ pi ≤ n) — the elements of the permutation p. Each of the last m lines contains two integers (aj, bj) (1 ≤ aj, bj ≤ n) — the pairs of positions to swap. Note that you are given a positions, not the values to swap. Output Print the only line with n distinct integers p'i (1 ≤ p'i ≤ n) — the lexicographically maximal permutation one can get. Example Input 9 6 1 2 3 4 5 6 7 8 9 1 4 4 7 2 5 5 8 3 6 6 9 Output 7 8 9 4 5 6 1 2 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. ZS the Coder has drawn an undirected graph of n vertices numbered from 0 to n - 1 and m edges between them. Each edge of the graph is weighted, each weight is a positive integer. The next day, ZS the Coder realized that some of the weights were erased! So he wants to reassign positive integer weight to each of the edges which weights were erased, so that the length of the shortest path between vertices s and t in the resulting graph is exactly L. Can you help him? Input The first line contains five integers n, m, L, s, t (2 ≤ n ≤ 1000, 1 ≤ m ≤ 10 000, 1 ≤ L ≤ 109, 0 ≤ s, t ≤ n - 1, s ≠ t) — the number of vertices, number of edges, the desired length of shortest path, starting vertex and ending vertex respectively. Then, m lines describing the edges of the graph follow. i-th of them contains three integers, ui, vi, wi (0 ≤ ui, vi ≤ n - 1, ui ≠ vi, 0 ≤ wi ≤ 109). ui and vi denote the endpoints of the edge and wi denotes its weight. If wi is equal to 0 then the weight of the corresponding edge was erased. It is guaranteed that there is at most one edge between any pair of vertices. Output Print "NO" (without quotes) in the only line if it's not possible to assign the weights in a required way. Otherwise, print "YES" in the first line. Next m lines should contain the edges of the resulting graph, with weights assigned to edges which weights were erased. i-th of them should contain three integers ui, vi and wi, denoting an edge between vertices ui and vi of weight wi. The edges of the new graph must coincide with the ones in the graph from the input. The weights that were not erased must remain unchanged whereas the new weights can be any positive integer not exceeding 1018. The order of the edges in the output doesn't matter. The length of the shortest path between s and t must be equal to L. If there are multiple solutions, print any of them. Examples Input 5 5 13 0 4 0 1 5 2 1 2 3 2 3 1 4 0 4 3 4 Output YES 0 1 5 2 1 2 3 2 3 1 4 8 4 3 4 Input 2 1 123456789 0 1 0 1 0 Output YES 0 1 123456789 Input 2 1 999999999 1 0 0 1 1000000000 Output NO Note Here's how the graph in the first sample case looks like : <image> In the first sample case, there is only one missing edge weight. Placing the weight of 8 gives a shortest path from 0 to 4 of length 13. In the second sample case, there is only a single edge. Clearly, the only way is to replace the missing weight with 123456789. In the last sample case, there is no weights to assign but the length of the shortest path doesn't match the required value, so the answer is "NO". 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 is currently at a car rental service, and he wants to reach cinema. The film he has bought a ticket for starts in t minutes. There is a straight road of length s from the service to the cinema. Let's introduce a coordinate system so that the car rental service is at the point 0, and the cinema is at the point s. There are k gas stations along the road, and at each of them you can fill a car with any amount of fuel for free! Consider that this operation doesn't take any time, i.e. is carried out instantly. There are n cars in the rental service, i-th of them is characterized with two integers ci and vi — the price of this car rent and the capacity of its fuel tank in liters. It's not allowed to fuel a car with more fuel than its tank capacity vi. All cars are completely fueled at the car rental service. Each of the cars can be driven in one of two speed modes: normal or accelerated. In the normal mode a car covers 1 kilometer in 2 minutes, and consumes 1 liter of fuel. In the accelerated mode a car covers 1 kilometer in 1 minutes, but consumes 2 liters of fuel. The driving mode can be changed at any moment and any number of times. Your task is to choose a car with minimum price such that Vasya can reach the cinema before the show starts, i.e. not later than in t minutes. Assume that all cars are completely fueled initially. Input The first line contains four positive integers n, k, s and t (1 ≤ n ≤ 2·105, 1 ≤ k ≤ 2·105, 2 ≤ s ≤ 109, 1 ≤ t ≤ 2·109) — the number of cars at the car rental service, the number of gas stations along the road, the length of the road and the time in which the film starts. Each of the next n lines contains two positive integers ci and vi (1 ≤ ci, vi ≤ 109) — the price of the i-th car and its fuel tank capacity. The next line contains k distinct integers g1, g2, ..., gk (1 ≤ gi ≤ s - 1) — the positions of the gas stations on the road in arbitrary order. Output Print the minimum rent price of an appropriate car, i.e. such car that Vasya will be able to reach the cinema before the film starts (not later than in t minutes). If there is no appropriate car, print -1. Examples Input 3 1 8 10 10 8 5 7 11 9 3 Output 10 Input 2 2 10 18 10 4 20 6 5 3 Output 20 Note In the first sample, Vasya can reach the cinema in time using the first or the third cars, but it would be cheaper to choose the first one. Its price is equal to 10, and the capacity of its fuel tank is 8. Then Vasya can drive to the first gas station in the accelerated mode in 3 minutes, spending 6 liters of fuel. After that he can full the tank and cover 2 kilometers in the normal mode in 4 minutes, spending 2 liters of fuel. Finally, he drives in the accelerated mode covering the remaining 3 kilometers in 3 minutes and spending 6 liters of fuel. 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. Programmers' kids solve this riddle in 5-10 minutes. How fast can you do it? Input The input contains a single integer n (0 ≤ n ≤ 2000000000). Output Output a single integer. Examples Input 11 Output 2 Input 14 Output 0 Input 61441 Output 2 Input 571576 Output 10 Input 2128506 Output 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. Arpa is taking a geometry exam. Here is the last problem of the exam. You are given three points a, b, c. Find a point and an angle such that if we rotate the page around the point by the angle, the new position of a is the same as the old position of b, and the new position of b is the same as the old position of c. Arpa is doubting if the problem has a solution or not (i.e. if there exists a point and an angle satisfying the condition). Help Arpa determine if the question has a solution or not. Input The only line contains six integers ax, ay, bx, by, cx, cy (\|ax\|, \|ay\|, \|bx\|, \|by\|, \|cx\|, \|cy\| ≤ 109). It's guaranteed that the points are distinct. Output Print "Yes" if the problem has a solution, "No" otherwise. You can print each letter in any case (upper or lower). Examples Input 0 1 1 1 1 0 Output Yes Input 1 1 0 0 1000 1000 Output No Note In the first sample test, rotate the page around (0.5, 0.5) by <image>. In the second sample test, you can't find any solution. 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. Disclaimer: there are lots of untranslateable puns in the Russian version of the statement, so there is one more reason for you to learn Russian :) Rick and Morty like to go to the ridge High Cry for crying loudly — there is an extraordinary echo. Recently they discovered an interesting acoustic characteristic of this ridge: if Rick and Morty begin crying simultaneously from different mountains, their cry would be heard between these mountains up to the height equal the bitwise OR of mountains they've climbed and all the mountains between them. Bitwise OR is a binary operation which is determined the following way. Consider representation of numbers x and y in binary numeric system (probably with leading zeroes) x = xk... x1x0 and y = yk... y1y0. Then z = x \| y is defined following way: z = zk... z1z0, where zi = 1, if xi = 1 or yi = 1, and zi = 0 otherwise. In the other words, digit of bitwise OR of two numbers equals zero if and only if digits at corresponding positions is both numbers equals zero. For example bitwise OR of numbers 10 = 10102 and 9 = 10012 equals 11 = 10112. In programming languages C/C++/Java/Python this operation is defined as «\|», and in Pascal as «or». Help Rick and Morty calculate the number of ways they can select two mountains in such a way that if they start crying from these mountains their cry will be heard above these mountains and all mountains between them. More formally you should find number of pairs l and r (1 ≤ l < r ≤ n) such that bitwise OR of heights of all mountains between l and r (inclusive) is larger than the height of any mountain at this interval. Input The first line contains integer n (1 ≤ n ≤ 200 000), the number of mountains in the ridge. Second line contains n integers ai (0 ≤ ai ≤ 109), the heights of mountains in order they are located in the ridge. Output Print the only integer, the number of ways to choose two different mountains. Examples Input 5 3 2 1 6 5 Output 8 Input 4 3 3 3 3 Output 0 Note In the first test case all the ways are pairs of mountains with the numbers (numbering from one): (1, 4), (1, 5), (2, 3), (2, 4), (2, 5), (3, 4), (3, 5), (4, 5) In the second test case there are no such pairs because for any pair of mountains the height of cry from them is 3, and this height is equal to the height of any mountain. 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 an undirected graph consisting of n vertices and <image> edges. Instead of giving you the edges that exist in the graph, we give you m unordered pairs (x, y) such that there is no edge between x and y, and if some pair of vertices is not listed in the input, then there is an edge between these vertices. You have to find the number of connected components in the graph and the size of each component. A connected component is a set of vertices X such that for every two vertices from this set there exists at least one path in the graph connecting these vertices, but adding any other vertex to X violates this rule. Input The first line contains two integers n and m (1 ≤ n ≤ 200000, <image>). Then m lines follow, each containing a pair of integers x and y (1 ≤ x, y ≤ n, x ≠ y) denoting that there is no edge between x and y. Each pair is listed at most once; (x, y) and (y, x) are considered the same (so they are never listed in the same test). If some pair of vertices is not listed in the input, then there exists an edge between those vertices. Output Firstly print k — the number of connected components in this graph. Then print k integers — the sizes of components. You should output these integers in non-descending order. Example Input 5 5 1 2 3 4 3 2 4 2 2 5 Output 2 1 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. Two-gram is an ordered pair (i.e. string of length two) of capital Latin letters. For example, "AZ", "AA", "ZA" — three distinct two-grams. You are given a string s consisting of n capital Latin letters. Your task is to find any two-gram contained in the given string as a substring (i.e. two consecutive characters of the string) maximal number of times. For example, for string s = "BBAABBBA" the answer is two-gram "BB", which contained in s three times. In other words, find any most frequent two-gram. Note that occurrences of the two-gram can overlap with each other. Input The first line of the input contains integer number n (2 ≤ n ≤ 100) — the length of string s. The second line of the input contains the string s consisting of n capital Latin letters. Output Print the only line containing exactly two capital Latin letters — any two-gram contained in the given string s as a substring (i.e. two consecutive characters of the string) maximal number of times. Examples Input 7 ABACABA Output AB Input 5 ZZZAA Output ZZ Note In the first example "BA" is also valid answer. In the second example the only two-gram "ZZ" can be printed because it contained in the string "ZZZAA" two times. 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. Allen and Bessie are playing a simple number game. They both know a function f: \{0, 1\}^n → R, i. e. the function takes n binary arguments and returns a real value. At the start of the game, the variables x_1, x_2, ..., x_n are all set to -1. Each round, with equal probability, one of Allen or Bessie gets to make a move. A move consists of picking an i such that x_i = -1 and either setting x_i → 0 or x_i → 1. After n rounds all variables are set, and the game value resolves to f(x_1, x_2, ..., x_n). Allen wants to maximize the game value, and Bessie wants to minimize it. Your goal is to help Allen and Bessie find the expected game value! They will play r+1 times though, so between each game, exactly one value of f changes. In other words, between rounds i and i+1 for 1 ≤ i ≤ r, f(z_1, ..., z_n) → g_i for some (z_1, ..., z_n) ∈ \{0, 1\}^n. You are to find the expected game value in the beginning and after each change. Input The first line contains two integers n and r (1 ≤ n ≤ 18, 0 ≤ r ≤ 2^{18}). The next line contains 2^n integers c_0, c_1, ..., c_{2^n-1} (0 ≤ c_i ≤ 10^9), denoting the initial values of f. More specifically, f(x_0, x_1, ..., x_{n-1}) = c_x, if x = \overline{x_{n-1} … x_0} in binary. Each of the next r lines contains two integers z and g (0 ≤ z ≤ 2^n - 1, 0 ≤ g ≤ 10^9). If z = \overline{z_{n-1} ... z_0} in binary, then this means to set f(z_0, ..., z_{n-1}) → g. Output Print r+1 lines, the i-th of which denotes the value of the game f during the i-th round. Your answer must have absolute or relative error within 10^{-6}. Formally, let your answer be a, and the jury's answer be b. Your answer is considered correct if \frac{\|a - b\|}{max{(1, \|b\|)}} ≤ 10^{-6}. Examples Input 2 2 0 1 2 3 2 5 0 4 Output 1.500000 2.250000 3.250000 Input 1 0 2 3 Output 2.500000 Input 2 0 1 1 1 1 Output 1.000000 Note Consider the second test case. If Allen goes first, he will set x_1 → 1, so the final value will be 3. If Bessie goes first, then she will set x_1 → 0 so the final value will be 2. Thus the answer is 2.5. In the third test case, the game value will always be 1 regardless of Allen and Bessie's play. 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 bloodiest battle of World War II has started. Germany and its allies have attacked the Soviet union army to gain control of the city of Stalingrad (now Volgograd) in the south-western Soviet Union.The war has become intensive and the soviet union's 64th army is fighting against the germans. General "Vasily Chuikov" is commanding the soviet union troops on a large triangular soviet army post located on the west bank of the Don River.General "Vasily Chuikov" very well knows the location of three corners of soviet army post.He is trying to detect any intruder inside the post.He is able to locate the position of enemy soldiers through his radar. Now if the location of any enemy soldier is known,it is required to tell whether the enemy is inside or outside the soviet army post. note: Any location is given by its coordinates(X,Y). INPUT: The first line contain number of soviet army post "T"."T" lines follows then.Each of the "T" lines contains three coordinates of soviet army post (X1,Y1),(X2,Y2),(X3,Y3) and the coordinates of enemy soldier (X4,Y4). OUTPUT: For each testcase print "OUTSIDE" If the enemy soldier is outside the triangular shaped soviet army post,if the enemy is inside the post, print "INSIDE". Also if the three known coordinates of soviet army post does not form a triangle,print "INSIDE". Constraints: 1 ≤ T ≤ 1000 -100 ≤ coordinates ≤ 100 SAMPLE INPUT 2 -1 0 1 0 0 1 3 3 -5 0 5 0 0 5 1 0SAMPLE OUTPUT OUTSIDE INSIDE Explanation In the first case,the enemy is outside the post. In the second case,the enemy is inside the post. 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. Professor Sharma gives the following problem to his students: given two integers X( ≥ 2) and Y( ≥ 2) and tells them to find the smallest positive integral exponent E such that the decimal expansion of X^E begins with Y. For example, if X = 8 and Y= 51, then X^3 = 512 begins with Y= 51, so E= 3. Professor Sharma has also announced that he is only interested in values of X such that X is not a power of 10. The professor has a proof that in this case, at least one value of E exists for any Y. now your task is to perform professor's theory and check his theory for different values of X and Y . Input : The first line contains the number of test cases N(0<N ≤ 9). For each test case, there is a single line containing the integers X and Y. Output : For each test case, print the case number, followed by a space and a colon, followed by a single space, followed by a single integer showing the value of the smallest exponent E. Constraints 1<T<10 2<X,Y ≤ 10^5 SAMPLE INPUT 2 5 156 16 40 SAMPLE OUTPUT Case 1: 6 Case 2: 3 Explanation Case 1: 55 = 255 =1255 = 6255 = 3125*5 = 15625 = 6 so after 6 turns we gets our answer cos 156 is present in 15625. Case 2: 1616 = 25616 = 4096 = 3 so after 3 turns we gets our answer cos 40 which is present in 4096 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. PowerShell had N natural numbers. He wanted to test Xenny's speed in finding the sum and difference of several numbers. He decided to ask Xenny several questions. In each question, he gave him two positive integers L and R. He asked him to find the sum of all integers from index L to index R (inclusive) and the difference of all integers from index R to index L (inclusive). (Numbers are 1-indexed) He asked Xenny Q such questions. Your task is to report the answer that Xenny gave to each of the Q questions. Input Format: First line contains 2 space-separated integers - N and Q. Second line contains N space-separated natural numbers. Q lines follow - each line contains 2 space-separated postiive integers L and R. Output Format: Output Q lines - the answer to each question. Each line should contain 2 space-separated integers - the required sum and difference. Constraints: 1 ≤ N ≤ 10^6 1 ≤ L ≤ R ≤ 10^6 1 ≤ Q ≤ 10^5 -1000000 ≤ Numbers ≤ 1000000 SAMPLE INPUT 4 1 1 2 3 4 1 3 SAMPLE OUTPUT 6 0 Explanation 1 + 2 + 3 = 6 3 - 2 - 1 = 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. Find the number of integers between 1 and N (inclusive) that contains exactly K non-zero digits when written in base ten. Constraints * 1 \leq N < 10^{100} * 1 \leq K \leq 3 Input Input is given from Standard Input in the following format: N K Output Print the count. Examples Input 100 1 Output 19 Input 25 2 Output 14 Input 314159 2 Output 937 Input 9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 3 Output 117879300 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. Takahashi went to an all-you-can-eat buffet with N kinds of dishes and ate all of them (Dish 1, Dish 2, \ldots, Dish N) once. The i-th dish (1 \leq i \leq N) he ate was Dish A_i. When he eats Dish i (1 \leq i \leq N), he gains B_i satisfaction points. Additionally, when he eats Dish i+1 just after eating Dish i (1 \leq i \leq N - 1), he gains C_i more satisfaction points. Find the sum of the satisfaction points he gained. Constraints * All values in input are integers. * 2 \leq N \leq 20 * 1 \leq A_i \leq N * A_1, A_2, ..., A_N are all different. * 1 \leq B_i \leq 50 * 1 \leq C_i \leq 50 Input Input is given from Standard Input in the following format: N A_1 A_2 ... A_N B_1 B_2 ... B_N C_1 C_2 ... C_{N-1} Output Print the sum of the satisfaction points Takahashi gained, as an integer. Examples Input 3 3 1 2 2 5 4 3 6 Output 14 Input 4 2 3 4 1 13 5 8 24 45 9 15 Output 74 Input 2 1 2 50 50 50 Output 150 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 integers a_1, a_2, ..., a_N not less than 1. The values of a_1, a_2, ..., a_N are not known, but it is known that a_1 \times a_2 \times ... \times a_N = P. Find the maximum possible greatest common divisor of a_1, a_2, ..., a_N. Constraints * 1 \leq N \leq 10^{12} * 1 \leq P \leq 10^{12} Input Input is given from Standard Input in the following format: N P Output Print the answer. Examples Input 3 24 Output 2 Input 5 1 Output 1 Input 1 111 Output 111 Input 4 972439611840 Output 206 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 an integer sequence A of length N and an integer K. You will perform the following operation on this sequence Q times: * Choose a contiguous subsequence of length K, then remove the smallest element among the K elements contained in the chosen subsequence (if there are multiple such elements, choose one of them as you like). Let X and Y be the values of the largest and smallest element removed in the Q operations. You would like X-Y to be as small as possible. Find the smallest possible value of X-Y when the Q operations are performed optimally. Constraints * 1 \leq N \leq 2000 * 1 \leq K \leq N * 1 \leq Q \leq N-K+1 * 1 \leq A_i \leq 10^9 * All values in input are integers. Input Input is given from Standard Input in the following format: N K Q A_1 A_2 ... A_N Output Print the smallest possible value of X-Y. Examples Input 5 3 2 4 3 1 5 2 Output 1 Input 10 1 6 1 1 2 3 5 8 13 21 34 55 Output 7 Input 11 7 5 24979445 861648772 623690081 433933447 476190629 262703497 211047202 971407775 628894325 731963982 822804784 Output 451211184 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. Joisino is planning to open a shop in a shopping street. Each of the five weekdays is divided into two periods, the morning and the evening. For each of those ten periods, a shop must be either open during the whole period, or closed during the whole period. Naturally, a shop must be open during at least one of those periods. There are already N stores in the street, numbered 1 through N. You are given information of the business hours of those shops, F_{i,j,k}. If F_{i,j,k}=1, Shop i is open during Period k on Day j (this notation is explained below); if F_{i,j,k}=0, Shop i is closed during that period. Here, the days of the week are denoted as follows. Monday: Day 1, Tuesday: Day 2, Wednesday: Day 3, Thursday: Day 4, Friday: Day 5. Also, the morning is denoted as Period 1, and the afternoon is denoted as Period 2. Let c_i be the number of periods during which both Shop i and Joisino's shop are open. Then, the profit of Joisino's shop will be P_{1,c_1}+P_{2,c_2}+...+P_{N,c_N}. Find the maximum possible profit of Joisino's shop when she decides whether her shop is open during each period, making sure that it is open during at least one period. Constraints * 1≤N≤100 * 0≤F_{i,j,k}≤1 * For every integer i such that 1≤i≤N, there exists at least one pair (j,k) such that F_{i,j,k}=1. * -10^7≤P_{i,j}≤10^7 * All input values are integers. Input Input is given from Standard Input in the following format: N F_{1,1,1} F_{1,1,2} ... F_{1,5,1} F_{1,5,2} : F_{N,1,1} F_{N,1,2} ... F_{N,5,1} F_{N,5,2} P_{1,0} ... P_{1,10} : P_{N,0} ... P_{N,10} Output Print the maximum possible profit of Joisino's shop. Examples Input 1 1 1 0 1 0 0 0 1 0 1 3 4 5 6 7 8 9 -2 -3 4 -2 Output 8 Input 2 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 -2 -2 -2 -2 -2 -1 -1 -1 -1 -1 0 -2 -2 -2 -2 -2 -1 -1 -1 -1 -1 Output -2 Input 3 1 1 1 1 1 1 0 0 1 1 0 1 0 1 1 1 1 0 1 0 1 0 1 1 0 1 0 1 0 1 -8 6 -2 -8 -8 4 8 7 -6 2 2 -9 2 0 1 7 -5 0 -2 -6 5 5 6 -6 7 -9 6 -5 8 0 -9 -7 -7 Output 23 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. Mole decided to live in an abandoned mine. The structure of the mine is represented by a simple connected undirected graph which consists of N vertices numbered 1 through N and M edges. The i-th edge connects Vertices a_i and b_i, and it costs c_i yen (the currency of Japan) to remove it. Mole would like to remove some of the edges so that there is exactly one path from Vertex 1 to Vertex N that does not visit the same vertex more than once. Find the minimum budget needed to achieve this. Constraints * 2 \leq N \leq 15 * N-1 \leq M \leq N(N-1)/2 * 1 \leq a_i, b_i \leq N * 1 \leq c_i \leq 10^{6} * There are neither multiple edges nor self-loops in the given graph. * The given graph is connected. Input Input is given from Standard Input in the following format: N M a_1 b_1 c_1 : a_M b_M c_M Output Print the answer. Examples Input 4 6 1 2 100 3 1 100 2 4 100 4 3 100 1 4 100 3 2 100 Output 200 Input 2 1 1 2 1 Output 0 Input 15 22 8 13 33418 14 15 55849 7 10 15207 4 6 64328 6 9 86902 15 7 46978 8 14 53526 1 2 8720 14 12 37748 8 3 61543 6 5 32425 4 11 20932 3 12 55123 8 2 45333 9 12 77796 3 9 71922 12 15 70793 2 4 25485 11 6 1436 2 7 81563 7 11 97843 3 1 40491 Output 133677 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 has decided to play a game, where the player runs a railway company. There are M+1 stations on Snuke Line, numbered 0 through M. A train on Snuke Line stops at station 0 and every d-th station thereafter, where d is a predetermined constant for each train. For example, if d = 3, the train stops at station 0, 3, 6, 9, and so forth. There are N kinds of souvenirs sold in areas around Snuke Line. The i-th kind of souvenirs can be purchased when the train stops at one of the following stations: stations l_i, l_i+1, l_i+2, ..., r_i. There are M values of d, the interval between two stops, for trains on Snuke Line: 1, 2, 3, ..., M. For each of these M values, find the number of the kinds of souvenirs that can be purchased if one takes a train with that value of d at station 0. Here, assume that it is not allowed to change trains. Constraints * 1 ≦ N ≦ 3 × 10^{5} * 1 ≦ M ≦ 10^{5} * 1 ≦ l_i ≦ r_i ≦ M Input The input is given from Standard Input in the following format: N M l_1 r_1 : l_{N} r_{N} Output Print the answer in M lines. The i-th line should contain the maximum number of the kinds of souvenirs that can be purchased if one takes a train stopping every i-th station. Examples Input 3 3 1 2 2 3 3 3 Output 3 2 2 Input 7 9 1 7 5 9 5 7 5 9 1 1 6 8 3 4 Output 7 6 6 5 4 5 5 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. We have a string X, which has an even number of characters. Half the characters are `S`, and the other half are `T`. Takahashi, who hates the string `ST`, will perform the following operation 10^{10000} times: * Among the occurrences of `ST` in X as (contiguous) substrings, remove the leftmost one. If there is no occurrence, do nothing. Find the eventual length of X. Constraints * 2 ≦ \|X\| ≦ 200,000 * The length of X is even. * Half the characters in X are `S`, and the other half are `T`. Input The input is given from Standard Input in the following format: X Output Print the eventual length of X. Examples Input TSTTSS Output 4 Input SSTTST Output 0 Input TSSTTTSS 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.
Solve the programming task below in a Python markdown code block. There is a 120 minute videotape with standard recording. When I set the VCR counter to 00:00:00 with the tape completely rewound and recorded in standard recording mode, I got a certain counter value. Enter this counter value (hours, minutes, seconds), find the length of the remaining tape (recordable time), and create a program that outputs in the format of hours: minutes: seconds. However, the input must be within 2 hours (120 minutes). The remaining amount of tape is calculated in two ways, standard recording mode and triple recording mode, and outputs two digits each for hours, minutes, and seconds as shown in the output example. If the tens digit is 0, such as "05", add "0". input Given multiple datasets. Each dataset is as follows. T H S T, H, and S are integers that represent hours, minutes, and seconds, respectively. Input ends when T, H, and S are all -1. The number of datasets does not exceed 50. output For each dataset On the first line, the hours, minutes, and seconds of the recordable time when the rest of the tape is recorded as standard, separated by half-width colons. On the second line, the hours, minutes, and seconds of the recordable time when the rest of the tape is recorded three times are separated by half-width colons. Please output. Example Input 1 30 0 -1 -1 -1 Output 00:30:00 01:30:00 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 want to go on a trip with a friend. However, friends who have a habit of spending money cannot easily save travel expenses. I don't know when my friends will go on a trip if they continue their current lives. So, if you want to travel early, you decide to create a program to help your friends save in a planned manner. If you have a friend's pocket money of M yen and the money you spend in that month is N yen, you will save (M --N) yen in that month. Create a program that inputs the monthly income and expenditure information M and N and outputs the number of months it takes for the savings amount to reach the travel cost L. However, if your savings do not reach your travel expenses after 12 months, print NA. 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: L M1 N1 M2 N2 :: M12 N12 The first line gives the travel cost L (1 ≤ L ≤ 1000000, integer). The next 12 lines are given the balance information for the i month, Mi, Ni (0 ≤ Mi, Ni ≤ 100000, Ni ≤ Mi, integer). The number of datasets does not exceed 1000. Output For each input dataset, print the number of months it takes for your savings to reach your travel costs on a single line. Example Input 10000 5000 3150 5000 5000 0 0 5000 1050 5000 3980 5000 210 5000 5000 5000 5000 0 0 5000 2100 5000 2100 5000 2100 29170 100000 100000 100000 100000 100000 100000 100000 100000 100000 100000 100000 100000 100000 100000 100000 100000 100000 100000 100000 100000 100000 100000 100000 70831 0 Output 6 NA 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 trick of fate caused Hatsumi and Taku to come to know each other. To keep the encounter in memory, they decided to calculate the difference between their ages. But the difference in ages varies depending on the day it is calculated. While trying again and again, they came to notice that the difference of their ages will hit a maximum value even though the months move on forever. Given the birthdays for the two, make a program to report the maximum difference between their ages. The age increases by one at the moment the birthday begins. If the birthday coincides with the 29th of February in a leap year, the age increases at the moment the 1st of March arrives in non-leap years. Input The input is given in the following format. y_1 m_1 d_1 y_2 m_2 d_2 The first and second lines provide Hatsumi’s and Taku’s birthdays respectively in year y_i (1 ≤ y_i ≤ 3000), month m_i (1 ≤ m_i ≤ 12), and day d_i (1 ≤ d_i ≤ Dmax) format. Where Dmax is given as follows: * 28 when February in a non-leap year * 29 when February in a leap-year * 30 in April, June, September, and November * 31 otherwise. It is a leap year if the year represented as a four-digit number is divisible by 4. Note, however, that it is a non-leap year if divisible by 100, and a leap year if divisible by 400. Output Output the maximum difference between their ages. Examples Input 1999 9 9 2001 11 3 Output 3 Input 2008 2 29 2015 3 1 Output 8 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 new type of mobile robot has been developed for environmental earth observation. It moves around on the ground, acquiring and recording various sorts of observational data using high precision sensors. Robots of this type have short range wireless communication devices and can exchange observational data with ones nearby. They also have large capacity memory units, on which they record data observed by themselves and those received from others. Figure 1 illustrates the current positions of three robots A, B, and C and the geographic coverage of their wireless devices. Each circle represents the wireless coverage of a robot, with its center representing the position of the robot. In this figure, two robots A and B are in the positions where A can transmit data to B, and vice versa. In contrast, C cannot communicate with A or B, since it is too remote from them. Still, however, once B moves towards C as in Figure 2, B and C can start communicating with each other. In this manner, B can relay observational data from A to C. Figure 3 shows another example, in which data propagate among several robots instantaneously. <image> --- Figure 1: The initial configuration of three robots <image> --- Figure 2: Mobile relaying <image> --- Figure 3: Instantaneous relaying among multiple robots As you may notice from these examples, if a team of robots move properly, observational data quickly spread over a large number of them. Your mission is to write a program that simulates how information spreads among robots. Suppose that, regardless of data size, the time necessary for communication is negligible. Input The input consists of multiple datasets, each in the following format. > N T R > nickname and travel route of the first robot > nickname and travel route of the second robot > ... > nickname and travel route of the N-th robot > The first line contains three integers N, T, and R that are the number of robots, the length of the simulation period, and the maximum distance wireless signals can reach, respectively, and satisfy that 1 <=N <= 100, 1 <= T <= 1000, and 1 <= R <= 10. The nickname and travel route of each robot are given in the following format. > nickname > t0 x0 y0 > t1 vx1 vy1 > t2 vx2 vy2 > ... > tk vxk vyk > Nickname is a character string of length between one and eight that only contains lowercase letters. No two robots in a dataset may have the same nickname. Each of the lines following nickname contains three integers, satisfying the following conditions. > 0 = t0 < t1 < ... < tk = T > -10 <= vx1, vy1, ..., vxk, vyk<= 10 > A robot moves around on a two dimensional plane. (x0, y0) is the location of the robot at time 0. From time ti-1 to ti (0 < i <= k), the velocities in the x and y directions are vxi and vyi, respectively. Therefore, the travel route of a robot is piecewise linear. Note that it may self-overlap or self-intersect. You may assume that each dataset satisfies the following conditions. * The distance between any two robots at time 0 is not exactly R. * The x- and y-coordinates of each robot are always between -500 and 500, inclusive. * Once any robot approaches within R + 10-6 of any other, the distance between them will become smaller than R - 10-6 while maintaining the velocities. * Once any robot moves away up to R - 10-6 of any other, the distance between them will become larger than R + 10-6 while maintaining the velocities. * If any pair of robots mutually enter the wireless area of the opposite ones at time t and any pair, which may share one or two members with the aforementioned pair, mutually leave the wireless area of the opposite ones at time t', the difference between t and t' is no smaller than 10-6 time unit, that is, \|t - t' \| >= 10-6. A dataset may include two or more robots that share the same location at the same time. However, you should still consider that they can move with the designated velocities. The end of the input is indicated by a line containing three zeros. Output For each dataset in the input, your program should print the nickname of each robot that have got until time T the observational data originally acquired by the first robot at time 0. Each nickname should be written in a separate line in dictionary order without any superfluous characters such as leading or trailing spaces. Example Input 3 5 10 red 0 0 0 5 0 0 green 0 5 5 5 6 1 blue 0 40 5 5 0 0 3 10 5 atom 0 47 32 5 -10 -7 10 1 0 pluto 0 0 0 7 0 0 10 3 3 gesicht 0 25 7 5 -7 -2 10 -1 10 4 100 7 impulse 0 -500 0 100 10 1 freedom 0 -491 0 100 9 2 destiny 0 -472 0 100 7 4 strike 0 -482 0 100 8 3 0 0 0 Output blue green red atom gesicht pluto freedom impulse strike 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 coordinates (a1, a2) and (b1, b2) on a two-dimensional grid of r × c are given. The cost of moving from a cell (e, f) to one of the cells (e + 1, f), (e-1, f), (e, f + 1), (e, f-1) is 1. And. You can also move between (e, c-1) and (e, 0), and between (r-1, f) and (0, f) at a cost of 1. At this time, find the number of routes that can be moved from the first coordinate to the second coordinate at the shortest cost. Input The input is given in the following format. r c a1 a2 b1 b2 Input meets the following constraints 1 ≤ r, c ≤ 1,000 0 ≤ a1, b1 <r 0 ≤ a2, b2 <c Output Output the remainder of the answer value divided by 100,000,007. Examples Input 4 4 0 0 3 3 Output 2 Input 4 4 0 0 1 1 Output 2 Input 2 3 0 0 1 2 Output 4 Input 500 500 0 0 200 200 Output 34807775 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. KM country has N kinds of coins and each coin has its value a_i. The king of the country, Kita_masa, thought that the current currency system is poor, and he decided to make it beautiful by changing the values of some (possibly no) coins. A currency system is called beautiful if each coin has an integer value and the (i+1)-th smallest value is divisible by the i-th smallest value for all i (1 \leq i \leq N-1). For example, the set {1, 5, 10, 50, 100, 500} is considered as a beautiful system, while the set {1, 5, 10, 25, 50, 100} is NOT, because 25 is not divisible by 10. Since changing the currency system may confuse citizens, the king, Kita_masa, wants to minimize the maximum value of the confusion ratios. Here, the confusion ratio for the change in the i-th coin is defined as \|a_i - b_i\| / a_i, where a_i and b_i is the value of i-th coin before and after the structure changes, respectively. Note that Kita_masa can change the value of each existing coin, but he cannot introduce new coins nor eliminate existing coins. After the modification, the values of two or more coins may coincide. Input Each dataset contains two lines. The first line contains a single integer, N, and the second line contains N integers, {a_i}. You may assume the following constraints: 1 \leq N \leq 20 1 \leq a_1 \lt a_2 \lt... \lt a_N \lt 10^5 Output Output one number that represents the minimum of the maximum value of the confusion ratios. The value may be printed with an arbitrary number of decimal digits, but may not contain an absolute error greater than or equal to 10^{-8}. Examples Input 3 6 11 12 Output 0.090909090909 Input 3 6 11 24 Output 0.090909090909 Input 3 6 11 30 Output 0.166666666667 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 given polygon g, print "1" if g is a convex polygon, "0" otherwise. Here, in a convex polygon, all interior angles are less than or equal to 180 degrees. g is represented by a sequence of points p1, p2,..., pn where line segments connecting pi and pi+1 (1 ≤ i ≤ n-1) are sides of the polygon. The line segment connecting pn and p1 is also a side of the polygon. Constraints * 3 ≤ n ≤ 100 * -10000 ≤ xi, yi ≤ 10000 * No point of the polygon will occur more than once. * Two sides of the polygon can intersect only at a common endpoint. Input g is given by coordinates of the points p1,..., pn in the following format: n x1 y1 x2 y2 : xn yn The first integer n is the number of points. The coordinate of a point pi is given by two integers xi and yi. The coordinates of points are given in the order of counter-clockwise visit of them. Output Print "1" or "0" in a line. Examples Input 4 0 0 3 1 2 3 0 3 Output 1 Input 5 0 0 2 0 1 1 2 2 0 2 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. Write a program which reads a sequence of integers $A = \\{a_0, a_1, ..., a_{n-1}\\}$ and swap specified elements by a list of the following operation: * swapRange($b, e, t$): For each integer $k$ ($0 \leq k < (e - b)$, swap element $(b + k)$ and element $(t + k)$. Constraints * $1 \leq n \leq 1,000$ * $-1,000,000,000 \leq a_i \leq 1,000,000,000$ * $1 \leq q \leq 1,000$ * $0 \leq b_i < e_i \leq n$ * $0 \leq t_i < t_i + (e_i - b_i) \leq n$ * Given swap ranges do not overlap each other Input The input is given in the following format. $n$ $a_0 \; a_1 \; ...,\; a_{n-1}$ $q$ $b_1 \; e_1 \; t_1$ $b_2 \; e_2 \; t_2$ : $b_{q} \; e_{q} \; t_{q}$ In the first line, $n$ (the number of elements in $A$) is given. In the second line, $a_i$ (each element in $A$) are given. In the third line, the number of queries $q$ is given and each query is given by three integers $b_i \; e_i \; t_i$ in the following $q$ lines. Output Print all elements of $A$ in a line after performing the given operations. Put a single space character between adjacency elements and a newline at the end of the last element. Example Input 11 1 2 3 4 5 6 7 8 9 10 11 1 1 4 7 Output 1 8 9 10 5 6 7 2 3 4 11 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. Innopolis University scientists continue to investigate the periodic table. There are n·m known elements and they form a periodic table: a rectangle with n rows and m columns. Each element can be described by its coordinates (r, c) (1 ≤ r ≤ n, 1 ≤ c ≤ m) in the table. Recently scientists discovered that for every four different elements in this table that form a rectangle with sides parallel to the sides of the table, if they have samples of three of the four elements, they can produce a sample of the fourth element using nuclear fusion. So if we have elements in positions (r1, c1), (r1, c2), (r2, c1), where r1 ≠ r2 and c1 ≠ c2, then we can produce element (r2, c2). <image> Samples used in fusion are not wasted and can be used again in future fusions. Newly crafted elements also can be used in future fusions. Innopolis University scientists already have samples of q elements. They want to obtain samples of all n·m elements. To achieve that, they will purchase some samples from other laboratories and then produce all remaining elements using an arbitrary number of nuclear fusions in some order. Help them to find the minimal number of elements they need to purchase. Input The first line contains three integers n, m, q (1 ≤ n, m ≤ 200 000; 0 ≤ q ≤ min(n·m, 200 000)), the chemical table dimensions and the number of elements scientists already have. The following q lines contain two integers ri, ci (1 ≤ ri ≤ n, 1 ≤ ci ≤ m), each describes an element that scientists already have. All elements in the input are different. Output Print the minimal number of elements to be purchased. Examples Input 2 2 3 1 2 2 2 2 1 Output 0 Input 1 5 3 1 3 1 1 1 5 Output 2 Input 4 3 6 1 2 1 3 2 2 2 3 3 1 3 3 Output 1 Note For each example you have a picture which illustrates it. The first picture for each example describes the initial set of element samples available. Black crosses represent elements available in the lab initially. The second picture describes how remaining samples can be obtained. Red dashed circles denote elements that should be purchased from other labs (the optimal solution should minimize the number of red circles). Blue dashed circles are elements that can be produced with nuclear fusion. They are numbered in order in which they can be produced. Test 1 We can use nuclear fusion and get the element from three other samples, so we don't need to purchase anything. <image> Test 2 We cannot use any nuclear fusion at all as there is only one row, so we have to purchase all missing elements. <image> Test 3 There are several possible solutions. One of them is illustrated below. Note that after purchasing one element marked as red it's still not possible to immidiately produce the middle element in the bottom row (marked as 4). So we produce the element in the left-top corner first (marked as 1), and then use it in future fusions. <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. You are given an array a of n integers and an integer s. It is guaranteed that n is odd. In one operation you can either increase or decrease any single element by one. Calculate the minimum number of operations required to make the median of the array being equal to s. The median of the array with odd length is the value of the element which is located on the middle position after the array is sorted. For example, the median of the array 6, 5, 8 is equal to 6, since if we sort this array we will get 5, 6, 8, and 6 is located on the middle position. Input The first line contains two integers n and s (1≤ n≤ 2⋅ 10^5-1, 1≤ s≤ 10^9) — the length of the array and the required value of median. The second line contains n integers a_1, a_2, …, a_n (1≤ a_i ≤ 10^9) — the elements of the array a. It is guaranteed that n is odd. Output In a single line output the minimum number of operations to make the median being equal to s. Examples Input 3 8 6 5 8 Output 2 Input 7 20 21 15 12 11 20 19 12 Output 6 Note In the first sample, 6 can be increased twice. The array will transform to 8, 5, 8, which becomes 5, 8, 8 after sorting, hence the median is equal to 8. In the second sample, 19 can be increased once and 15 can be increased five times. The array will become equal to 21, 20, 12, 11, 20, 20, 12. If we sort this array we get 11, 12, 12, 20, 20, 20, 21, this way the median is 20. 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. Each item in the game has a level. The higher the level is, the higher basic parameters the item has. We shall consider only the following basic parameters: attack (atk), defense (def) and resistance to different types of impact (res). Each item belongs to one class. In this problem we will only consider three of such classes: weapon, armor, orb. Besides, there's a whole new world hidden inside each item. We can increase an item's level travelling to its world. We can also capture the so-called residents in the Item World Residents are the creatures that live inside items. Each resident gives some bonus to the item in which it is currently located. We will only consider residents of types: gladiator (who improves the item's atk), sentry (who improves def) and physician (who improves res). Each item has the size parameter. The parameter limits the maximum number of residents that can live inside an item. We can move residents between items. Within one moment of time we can take some resident from an item and move it to some other item if it has a free place for a new resident. We cannot remove a resident from the items and leave outside — any of them should be inside of some item at any moment of time. Laharl has a certain number of items. He wants to move the residents between items so as to equip himself with weapon, armor and a defensive orb. The weapon's atk should be largest possible in the end. Among all equipping patterns containing weapon's maximum atk parameter we should choose the ones where the armor’s def parameter is the largest possible. Among all such equipment patterns we should choose the one where the defensive orb would have the largest possible res parameter. Values of the parameters def and res of weapon, atk and res of armor and atk and def of orb are indifferent for Laharl. Find the optimal equipment pattern Laharl can get. Input The first line contains number n (3 ≤ n ≤ 100) — representing how many items Laharl has. Then follow n lines. Each line contains description of an item. The description has the following form: "name class atk def res size" — the item's name, class, basic attack, defense and resistance parameters and its size correspondingly. * name and class are strings and atk, def, res and size are integers. * name consists of lowercase Latin letters and its length can range from 1 to 10, inclusive. * class can be "weapon", "armor" or "orb". * 0 ≤ atk, def, res ≤ 1000. * 1 ≤ size ≤ 10. It is guaranteed that Laharl has at least one item of each class. The next line contains an integer k (1 ≤ k ≤ 1000) — the number of residents. Then k lines follow. Each of them describes a resident. A resident description looks like: "name type bonus home" — the resident's name, his type, the number of points the resident adds to the item's corresponding parameter and the name of the item which currently contains the resident. * name, type and home are strings and bonus is an integer. * name consists of lowercase Latin letters and its length can range from 1 to 10, inclusive. * type may be "gladiator", "sentry" or "physician". * 1 ≤ bonus ≤ 100. It is guaranteed that the number of residents in each item does not exceed the item's size. The names of all items and residents are pairwise different. All words and numbers in the input are separated by single spaces. Output Print on the first line the name of the weapon in the optimal equipping pattern; then print the number of residents the weapon contains; then print the residents' names. Print on the second and third lines in the same form the names of the armor and defensive orb as well as the residents they contain. Use single spaces for separation. If there are several possible solutions, print any of them. Examples Input 4 sword weapon 10 2 3 2 pagstarmor armor 0 15 3 1 iceorb orb 3 2 13 2 longbow weapon 9 1 2 1 5 mike gladiator 5 longbow bobby sentry 6 pagstarmor petr gladiator 7 iceorb teddy physician 6 sword blackjack sentry 8 sword Output sword 2 petr mike pagstarmor 1 blackjack iceorb 2 teddy bobby Input 4 sword weapon 10 2 3 2 pagstarmor armor 0 15 3 1 iceorb orb 3 2 13 2 longbow weapon 9 1 2 1 6 mike gladiator 5 longbow bobby sentry 6 pagstarmor petr gladiator 7 iceorb teddy physician 6 sword blackjack sentry 8 sword joe physician 6 iceorb Output longbow 1 mike pagstarmor 1 bobby iceorb 2 petr joe Note In the second sample we have no free space inside the items, therefore we cannot move the residents between them. 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. Chouti was tired of the tedious homework, so he opened up an old programming problem he created years ago. You are given a connected undirected graph with n vertices and m weighted edges. There are k special vertices: x_1, x_2, …, x_k. Let's define the cost of the path as the maximum weight of the edges in it. And the distance between two vertexes as the minimum cost of the paths connecting them. For each special vertex, find another special vertex which is farthest from it (in terms of the previous paragraph, i.e. the corresponding distance is maximum possible) and output the distance between them. The original constraints are really small so he thought the problem was boring. Now, he raises the constraints and hopes you can solve it for him. Input The first line contains three integers n, m and k (2 ≤ k ≤ n ≤ 10^5, n-1 ≤ m ≤ 10^5) — the number of vertices, the number of edges and the number of special vertices. The second line contains k distinct integers x_1, x_2, …, x_k (1 ≤ x_i ≤ n). Each of the following m lines contains three integers u, v and w (1 ≤ u,v ≤ n, 1 ≤ w ≤ 10^9), denoting there is an edge between u and v of weight w. The given graph is undirected, so an edge (u, v) can be used in the both directions. The graph may have multiple edges and self-loops. It is guaranteed, that the graph is connected. Output The first and only line should contain k integers. The i-th integer is the distance between x_i and the farthest special vertex from it. Examples Input 2 3 2 2 1 1 2 3 1 2 2 2 2 1 Output 2 2 Input 4 5 3 1 2 3 1 2 5 4 2 1 2 3 2 1 4 4 1 3 3 Output 3 3 3 Note In the first example, the distance between vertex 1 and 2 equals to 2 because one can walk through the edge of weight 2 connecting them. So the distance to the farthest node for both 1 and 2 equals to 2. In the second example, one can find that distance between 1 and 2, distance between 1 and 3 are both 3 and the distance between 2 and 3 is 2. The graph may have multiple edges between and self-loops, as in the first 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. During the archaeological research in the Middle East you found the traces of three ancient religions: First religion, Second religion and Third religion. You compiled the information on the evolution of each of these beliefs, and you now wonder if the followers of each religion could coexist in peace. The Word of Universe is a long word containing the lowercase English characters only. At each moment of time, each of the religion beliefs could be described by a word consisting of lowercase English characters. The three religions can coexist in peace if their descriptions form disjoint subsequences of the Word of Universe. More formally, one can paint some of the characters of the Word of Universe in three colors: 1, 2, 3, so that each character is painted in at most one color, and the description of the i-th religion can be constructed from the Word of Universe by removing all characters that aren't painted in color i. The religions however evolve. In the beginning, each religion description is empty. Every once in a while, either a character is appended to the end of the description of a single religion, or the last character is dropped from the description. After each change, determine if the religions could coexist in peace. Input The first line of the input contains two integers n, q (1 ≤ n ≤ 100 000, 1 ≤ q ≤ 1000) — the length of the Word of Universe and the number of religion evolutions, respectively. The following line contains the Word of Universe — a string of length n consisting of lowercase English characters. Each of the following line describes a single evolution and is in one of the following formats: * + i c (i ∈ \{1, 2, 3\}, c ∈ \{a, b, ..., z\}: append the character c to the end of i-th religion description. * - i (i ∈ \{1, 2, 3\}) – remove the last character from the i-th religion description. You can assume that the pattern is non-empty. You can assume that no religion will have description longer than 250 characters. Output Write q lines. The i-th of them should be YES if the religions could coexist in peace after the i-th evolution, or NO otherwise. You can print each character in any case (either upper or lower). Examples Input 6 8 abdabc + 1 a + 1 d + 2 b + 2 c + 3 a + 3 b + 1 c - 2 Output YES YES YES YES YES YES NO YES Input 6 8 abbaab + 1 a + 2 a + 3 a + 1 b + 2 b + 3 b - 1 + 2 z Output YES YES YES YES YES NO YES NO Note In the first example, after the 6th evolution the religion descriptions are: ad, bc, and ab. The following figure shows how these descriptions form three disjoint subsequences of the Word of Universe: <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. Cengiz recently learned Fibonacci numbers and now he is studying different algorithms to find them. After getting bored of reading them, he came with his own new type of numbers that he named XORinacci numbers. He defined them as follows: * f(0) = a; * f(1) = b; * f(n) = f(n-1) ⊕ f(n-2) when n > 1, where ⊕ denotes the [bitwise XOR operation](https://en.wikipedia.org/wiki/Bitwise_operation#XOR). You are given three integers a, b, and n, calculate f(n). You have to answer for T independent test cases. Input The input contains one or more independent test cases. The first line of input contains a single integer T (1 ≤ T ≤ 10^3), the number of test cases. Each of the T following lines contains three space-separated integers a, b, and n (0 ≤ a, b, n ≤ 10^9) respectively. Output For each test case, output f(n). Example Input 3 3 4 2 4 5 0 325 265 1231232 Output 7 4 76 Note In the first example, f(2) = f(0) ⊕ f(1) = 3 ⊕ 4 = 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. Your math teacher gave you the following problem: There are n segments on the x-axis, [l_1; r_1], [l_2; r_2], …, [l_n; r_n]. The segment [l; r] includes the bounds, i.e. it is a set of such x that l ≤ x ≤ r. The length of the segment [l; r] is equal to r - l. Two segments [a; b] and [c; d] have a common point (intersect) if there exists x that a ≤ x ≤ b and c ≤ x ≤ d. For example, [2; 5] and [3; 10] have a common point, but [5; 6] and [1; 4] don't have. You should add one segment, which has at least one common point with each of the given segments and as short as possible (i.e. has minimal length). The required segment can degenerate to be a point (i.e a segment with length zero). The added segment may or may not be among the given n segments. In other words, you need to find a segment [a; b], such that [a; b] and every [l_i; r_i] have a common point for each i, and b-a is minimal. Input The first line contains integer number t (1 ≤ t ≤ 100) — the number of test cases in the input. Then t test cases follow. The first line of each test case contains one integer n (1 ≤ n ≤ 10^{5}) — the number of segments. The following n lines contain segment descriptions: the i-th of them contains two integers l_i,r_i (1 ≤ l_i ≤ r_i ≤ 10^{9}). The sum of all values n over all the test cases in the input doesn't exceed 10^5. Output For each test case, output one integer — the smallest possible length of the segment which has at least one common point with all given segments. Example Input 4 3 4 5 5 9 7 7 5 11 19 4 17 16 16 3 12 14 17 1 1 10 1 1 1 Output 2 4 0 0 Note In the first test case of the example, we can choose the segment [5;7] as the answer. It is the shortest segment that has at least one common point with all given segments. 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're given an array a_1, …, a_n of n non-negative integers. Let's call it sharpened if and only if there exists an integer 1 ≤ k ≤ n such that a_1 < a_2 < … < a_k and a_k > a_{k+1} > … > a_n. In particular, any strictly increasing or strictly decreasing array is sharpened. For example: * The arrays [4], [0, 1], [12, 10, 8] and [3, 11, 15, 9, 7, 4] are sharpened; * The arrays [2, 8, 2, 8, 6, 5], [0, 1, 1, 0] and [2, 5, 6, 9, 8, 8] are not sharpened. You can do the following operation as many times as you want: choose any strictly positive element of the array, and decrease it by one. Formally, you can choose any i (1 ≤ i ≤ n) such that a_i>0 and assign a_i := a_i - 1. Tell if it's possible to make the given array sharpened using some number (possibly zero) of these operations. Input The input consists of multiple test cases. The first line contains a single integer t (1 ≤ t ≤ 15\ 000) — the number of test cases. The description of the test cases follows. The first line of each test case contains a single integer n (1 ≤ n ≤ 3 ⋅ 10^5). The second line of each test case contains a sequence of n non-negative integers a_1, …, a_n (0 ≤ a_i ≤ 10^9). It is guaranteed that the sum of n over all test cases does not exceed 3 ⋅ 10^5. Output For each test case, output a single line containing "Yes" (without quotes) if it's possible to make the given array sharpened using the described operations, or "No" (without quotes) otherwise. Example Input 10 1 248618 3 12 10 8 6 100 11 15 9 7 8 4 0 1 1 0 2 0 0 2 0 1 2 1 0 2 1 1 3 0 1 0 3 1 0 1 Output Yes Yes Yes No No Yes Yes Yes Yes No Note In the first and the second test case of the first test, the given array is already sharpened. In the third test case of the first test, we can transform the array into [3, 11, 15, 9, 7, 4] (decrease the first element 97 times and decrease the last element 4 times). It is sharpened because 3 < 11 < 15 and 15 > 9 > 7 > 4. In the fourth test case of the first test, it's impossible to make the given array sharpened. 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 want to perform the combo on your opponent in one popular fighting game. The combo is the string s consisting of n lowercase Latin letters. To perform the combo, you have to press all buttons in the order they appear in s. I.e. if s="abca" then you have to press 'a', then 'b', 'c' and 'a' again. You know that you will spend m wrong tries to perform the combo and during the i-th try you will make a mistake right after p_i-th button (1 ≤ p_i < n) (i.e. you will press first p_i buttons right and start performing the combo from the beginning). It is guaranteed that during the m+1-th try you press all buttons right and finally perform the combo. I.e. if s="abca", m=2 and p = [1, 3] then the sequence of pressed buttons will be 'a' (here you're making a mistake and start performing the combo from the beginning), 'a', 'b', 'c', (here you're making a mistake and start performing the combo from the beginning), 'a' (note that at this point you will not perform the combo because of the mistake), 'b', 'c', 'a'. Your task is to calculate for each button (letter) the number of times you'll press it. You have to answer t independent test cases. Input The first line of the input contains one integer t (1 ≤ t ≤ 10^4) — the number of test cases. Then t test cases follow. The first line of each test case contains two integers n and m (2 ≤ n ≤ 2 ⋅ 10^5, 1 ≤ m ≤ 2 ⋅ 10^5) — the length of s and the number of tries correspondingly. The second line of each test case contains the string s consisting of n lowercase Latin letters. The third line of each test case contains m integers p_1, p_2, ..., p_m (1 ≤ p_i < n) — the number of characters pressed right during the i-th try. It is guaranteed that the sum of n and the sum of m both does not exceed 2 ⋅ 10^5 (∑ n ≤ 2 ⋅ 10^5, ∑ m ≤ 2 ⋅ 10^5). It is guaranteed that the answer for each letter does not exceed 2 ⋅ 10^9. Output For each test case, print the answer — 26 integers: the number of times you press the button 'a', the number of times you press the button 'b', ..., the number of times you press the button 'z'. Example Input 3 4 2 abca 1 3 10 5 codeforces 2 8 3 2 9 26 10 qwertyuioplkjhgfdsazxcvbnm 20 10 1 2 3 5 10 5 9 4 Output 4 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 9 4 5 3 0 0 0 0 0 0 0 0 9 0 0 3 1 0 0 0 0 0 0 0 2 1 1 2 9 2 2 2 5 2 2 2 1 1 5 4 11 8 2 7 5 1 10 1 5 2 Note The first test case is described in the problem statement. Wrong tries are "a", "abc" and the final try is "abca". The number of times you press 'a' is 4, 'b' is 2 and 'c' is 2. In the second test case, there are five wrong tries: "co", "codeforc", "cod", "co", "codeforce" and the final try is "codeforces". The number of times you press 'c' is 9, 'd' is 4, 'e' is 5, 'f' is 3, 'o' is 9, 'r' is 3 and 's' 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. Polycarp has recently created a new level in this cool new game Berlio Maker 85 and uploaded it online. Now players from all over the world can try his level. All levels in this game have two stats to them: the number of plays and the number of clears. So when a player attempts the level, the number of plays increases by 1. If he manages to finish the level successfully then the number of clears increases by 1 as well. Note that both of the statistics update at the same time (so if the player finishes the level successfully then the number of plays will increase at the same time as the number of clears). Polycarp is very excited about his level, so he keeps peeking at the stats to know how hard his level turns out to be. So he peeked at the stats n times and wrote down n pairs of integers — (p_1, c_1), (p_2, c_2), ..., (p_n, c_n), where p_i is the number of plays at the i-th moment of time and c_i is the number of clears at the same moment of time. The stats are given in chronological order (i.e. the order of given pairs is exactly the same as Polycarp has written down). Between two consecutive moments of time Polycarp peeked at the stats many players (but possibly zero) could attempt the level. Finally, Polycarp wonders if he hasn't messed up any records and all the pairs are correct. If there could exist such a sequence of plays (and clears, respectively) that the stats were exactly as Polycarp has written down, then he considers his records correct. Help him to check the correctness of his records. For your convenience you have to answer multiple independent test cases. Input The first line contains a single integer T (1 ≤ T ≤ 500) — the number of test cases. The first line of each test case contains a single integer n (1 ≤ n ≤ 100) — the number of moments of time Polycarp peeked at the stats. Each of the next n lines contains two integers p_i and c_i (0 ≤ p_i, c_i ≤ 1000) — the number of plays and the number of clears of the level at the i-th moment of time. Note that the stats are given in chronological order. Output For each test case print a single line. If there could exist such a sequence of plays (and clears, respectively) that the stats were exactly as Polycarp has written down, then print "YES". Otherwise, print "NO". You can print each letter in any case (upper or lower). Example Input 6 3 0 0 1 1 1 2 2 1 0 1000 3 4 10 1 15 2 10 2 15 2 1 765 432 2 4 4 4 3 5 0 0 1 0 1 0 1 0 1 0 Output NO YES NO YES NO YES Note In the first test case at the third moment of time the number of clears increased but the number of plays did not, that couldn't have happened. The second test case is a nice example of a Super Expert level. In the third test case the number of plays decreased, which is impossible. The fourth test case is probably an auto level with a single jump over the spike. In the fifth test case the number of clears decreased, which is also impossible. Nobody wanted to play the sixth test case; Polycarp's mom attempted it to make him feel better, however, she couldn't clear it. 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 statement of this problem is the same as the statement of problem C1. The only difference is that, in problem C1, n is always even, and in C2, n is always odd. You are given a regular polygon with 2 ⋅ n vertices (it's convex and has equal sides and equal angles) and all its sides have length 1. Let's name it as 2n-gon. Your task is to find the square of the minimum size such that you can embed 2n-gon in the square. Embedding 2n-gon in the square means that you need to place 2n-gon in the square in such way that each point which lies inside or on a border of 2n-gon should also lie inside or on a border of the square. You can rotate 2n-gon and/or the square. Input The first line contains a single integer T (1 ≤ T ≤ 200) — the number of test cases. Next T lines contain descriptions of test cases — one per line. Each line contains single odd integer n (3 ≤ n ≤ 199). Don't forget you need to embed 2n-gon, not an n-gon. Output Print T real numbers — one per test case. For each test case, print the minimum length of a side of the square 2n-gon can be embedded in. Your answer will be considered correct if its absolute or relative error doesn't exceed 10^{-6}. Example Input 3 3 5 199 Output 1.931851653 3.196226611 126.687663595 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 an integer n. In one move, you can either multiply n by two or divide n by 6 (if it is divisible by 6 without the remainder). Your task is to find the minimum number of moves needed to obtain 1 from n or determine if it's impossible to do that. You have to answer t independent test cases. Input The first line of the input contains one integer t (1 ≤ t ≤ 2 ⋅ 10^4) — 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). Output For each test case, print the answer — the minimum number of moves needed to obtain 1 from n if it's possible to do that or -1 if it's impossible to obtain 1 from n. Example Input 7 1 2 3 12 12345 15116544 387420489 Output 0 -1 2 -1 -1 12 36 Note Consider the sixth test case of the example. The answer can be obtained by the following sequence of moves from the given integer 15116544: 1. Divide by 6 and get 2519424; 2. divide by 6 and get 419904; 3. divide by 6 and get 69984; 4. divide by 6 and get 11664; 5. multiply by 2 and get 23328; 6. divide by 6 and get 3888; 7. divide by 6 and get 648; 8. divide by 6 and get 108; 9. multiply by 2 and get 216; 10. divide by 6 and get 36; 11. divide by 6 and get 6; 12. divide by 6 and get 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. You are given an array a of n integers. You want to make all elements of a equal to zero by doing the following operation exactly three times: * Select a segment, for each number in this segment we can add a multiple of len to it, where len is the length of this segment (added integers can be different). It can be proven that it is always possible to make all elements of a equal to zero. Input The first line contains one integer n (1 ≤ n ≤ 100 000): the number of elements of the array. The second line contains n elements of an array a separated by spaces: a_1, a_2, ..., a_n (-10^9 ≤ a_i ≤ 10^9). Output The output should contain six lines representing three operations. For each operation, print two lines: * The first line contains two integers l, r (1 ≤ l ≤ r ≤ n): the bounds of the selected segment. * The second line contains r-l+1 integers b_l, b_{l+1}, ..., b_r (-10^{18} ≤ b_i ≤ 10^{18}): the numbers to add to a_l, a_{l+1}, …, a_r, respectively; b_i should be divisible by r - l + 1. Example Input 4 1 3 2 4 Output 1 1 -1 3 4 4 2 2 4 -3 -6 -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. Ori and Sein have overcome many difficult challenges. They finally lit the Shrouded Lantern and found Gumon Seal, the key to the Forlorn Ruins. When they tried to open the door to the ruins... nothing happened. Ori was very surprised, but Sein gave the explanation quickly: clever Gumon decided to make an additional defence for the door. There are n lamps with Spirit Tree's light. Sein knows the time of turning on and off for the i-th lamp — l_i and r_i respectively. To open the door you have to choose k lamps in such a way that there will be a moment of time when they all will be turned on. While Sein decides which of the k lamps to pick, Ori is interested: how many ways there are to pick such k lamps that the door will open? It may happen that Sein may be wrong and there are no such k lamps. The answer might be large, so print it modulo 998 244 353. Input First line contains two integers n and k (1 ≤ n ≤ 3 ⋅ 10^5, 1 ≤ k ≤ n) — total number of lamps and the number of lamps that must be turned on simultaneously. Next n lines contain two integers l_i ans r_i (1 ≤ l_i ≤ r_i ≤ 10^9) — period of time when i-th lamp is turned on. Output Print one integer — the answer to the task modulo 998 244 353. Examples Input 7 3 1 7 3 8 4 5 6 7 1 3 5 10 8 9 Output 9 Input 3 1 1 1 2 2 3 3 Output 3 Input 3 2 1 1 2 2 3 3 Output 0 Input 3 3 1 3 2 3 3 3 Output 1 Input 5 2 1 3 2 4 3 5 4 6 5 7 Output 7 Note In first test case there are nine sets of k lamps: (1, 2, 3), (1, 2, 4), (1, 2, 5), (1, 2, 6), (1, 3, 6), (1, 4, 6), (2, 3, 6), (2, 4, 6), (2, 6, 7). In second test case k=1, so the answer is 3. In third test case there are no such pairs of lamps. In forth test case all lamps are turned on in a time 3, so the answer is 1. In fifth test case there are seven sets of k lamps: (1, 2), (1, 3), (2, 3), (2, 4), (3, 4), (3, 5), (4, 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. After his wife's tragic death, Eurydice, Orpheus descended to the realm of death to see her. Reaching its gates was uneasy, but passing through them proved to be even more challenging. Mostly because of Cerberus, the three-headed hound of Hades. Orpheus, a famous poet, and musician plans to calm Cerberus with his poetry and safely walk past him. He created a very peculiar poem for Cerberus. It consists only of lowercase English letters. We call a poem's substring a palindrome if and only if it reads the same backwards and forwards. A string a is a substring of a string b if a can be obtained from b by deleting several (possibly zero or all) characters from the beginning and several (possibly zero or all) characters from the end. Unfortunately, Cerberus dislikes palindromes of length greater than 1. For example in the poem abaa the hound of Hades wouldn't like substrings aba and aa. Orpheus can only calm Cerberus if the hound likes his poetry. That's why he wants to change his poem so that it does not contain any palindrome substrings of length greater than 1. Orpheus can modify the poem by replacing a letter at any position with any lowercase English letter. He can use this operation arbitrarily many times (possibly zero). Since there can be many palindromes in his poem, he may have to make some corrections. But how many, exactly? Given the poem, determine the minimal number of letters that have to be changed so that the poem does not contain any palindromes of length greater than 1. Input The first line of the input contains a single integer t (1 ≤ t ≤ 10^5) denoting the number of test cases, then t test cases follow. The first and only line of each test case contains a non-empty string of lowercase English letters, Orpheus' poem. The sum of the length of Orpheus' poems in all test cases will not exceed 10^5. Output You should output t lines, i-th line should contain a single integer, answer to the i-th test case. Example Input 7 babba abaac codeforces zeroorez abcdcba bbbbbbb a Output 1 1 0 1 1 4 0 Note In the first test case, we can replace the third character with c and obtain a palindrome-less poem bacba. In the second test case, we can replace the third character with d and obtain a palindrome-less poem abdac. In the third test case, the initial poem already doesn't contain any palindromes, so Orpheus doesn't need to change anything there. 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 — is a sequence of length n integers from 1 to n, in which all the numbers occur exactly once. For example, [1], [3, 5, 2, 1, 4], [1, 3, 2] — permutations, and [2, 3, 2], [4, 3, 1], [0] — no. Polycarp was recently gifted a permutation a[1 ... n] of length n. Polycarp likes trees more than permutations, so he wants to transform permutation a into a rooted binary tree. He transforms an array of different integers into a tree as follows: * the maximum element of the array becomes the root of the tree; * all elements to the left of the maximum — form a left subtree (which is built according to the same rules but applied to the left part of the array), but if there are no elements to the left of the maximum, then the root has no left child; * all elements to the right of the maximum — form a right subtree (which is built according to the same rules but applied to the right side of the array), but if there are no elements to the right of the maximum, then the root has no right child. For example, if he builds a tree by permutation a=[3, 5, 2, 1, 4], then the root will be the element a_2=5, and the left subtree will be the tree that will be built for the subarray a[1 ... 1] = [3], and the right one — for the subarray a[3 ... 5] = [2, 1, 4]. As a result, the following tree will be built: <image> The tree corresponding to the permutation a=[3, 5, 2, 1, 4]. Another example: let the permutation be a=[1, 3, 2, 7, 5, 6, 4]. In this case, the tree looks like this: <image> The tree corresponding to the permutation a=[1, 3, 2, 7, 5, 6, 4]. Let us denote by d_v the depth of the vertex a_v, that is, the number of edges on the path from the root to the vertex numbered a_v. Note that the root depth is zero. Given the permutation a, for each vertex, find the value of d_v. Input The first line contains one integer t (1 ≤ t ≤ 100) — the number of test cases. Then t test cases follow. The first line of each test case contains an integer n (1 ≤ n ≤ 100) — the length of the permutation. This is followed by n numbers a_1, a_2, …, a_n — permutation a. Output For each test case, output n values — d_1, d_2, …, d_n. Example Input 3 5 3 5 2 1 4 1 1 4 4 3 1 2 Output 1 0 2 3 1 0 0 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. In some country live wizards. They love playing with numbers. The blackboard has two numbers written on it — a and b. The order of the numbers is not important. Let's consider a ≤ b for the sake of definiteness. The players can cast one of the two spells in turns: * Replace b with b - ak. Number k can be chosen by the player, considering the limitations that k > 0 and b - ak ≥ 0. Number k is chosen independently each time an active player casts a spell. * Replace b with b mod a. If a > b, similar moves are possible. If at least one of the numbers equals zero, a player can't make a move, because taking a remainder modulo zero is considered somewhat uncivilized, and it is far too boring to subtract a zero. The player who cannot make a move, loses. To perform well in the magic totalizator, you need to learn to quickly determine which player wins, if both wizards play optimally: the one that moves first or the one that moves second. Input The first line contains a single integer t — the number of input data sets (1 ≤ t ≤ 104). Each of the next t lines contains two integers a, b (0 ≤ a, b ≤ 1018). The numbers are separated by a space. Please do not use the %lld specificator to read or write 64-bit integers in С++. It is preferred to use the cin, cout streams or the %I64d specificator. Output For any of the t input sets print "First" (without the quotes) if the player who moves first wins. Print "Second" (without the quotes) if the player who moves second wins. Print the answers to different data sets on different lines in the order in which they are given in the input. Examples Input 4 10 21 31 10 0 1 10 30 Output First Second Second First Note In the first sample, the first player should go to (11,10). Then, after a single move of the second player to (1,10), he will take 10 modulo 1 and win. In the second sample the first player has two moves to (1,10) and (21,10). After both moves the second player can win. In the third sample, the first player has no moves. In the fourth sample, the first player wins in one move, taking 30 modulo 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. Flatland is inhabited by pixels of three colors: red, green and blue. We know that if two pixels of different colors meet in a violent fight, only one of them survives the fight (that is, the total number of pixels decreases by one). Besides, if pixels of colors x and y (x ≠ y) meet in a violent fight, then the pixel that survives the fight immediately changes its color to z (z ≠ x; z ≠ y). Pixels of the same color are friends, so they don't fight. The King of Flatland knows that his land will be peaceful and prosperous when the pixels are of the same color. For each of the three colors you know the number of pixels of this color that inhabit Flatland. Help the king and determine whether fights can bring peace and prosperity to the country and if it is possible, find the minimum number of fights needed to make the land peaceful and prosperous. Input The first line contains three space-separated integers a, b and c (0 ≤ a, b, c ≤ 231; a + b + c > 0) — the number of red, green and blue pixels, correspondingly. Output Print a single number — the minimum number of pixel fights before the country becomes peaceful and prosperous. If making the country peaceful and prosperous is impossible, print -1. Examples Input 1 1 1 Output 1 Input 3 1 0 Output 3 Note In the first test sample the country needs only one fight to achieve peace and prosperity. Besides, it can be any fight whatsoever. For example, let's assume that the green and the blue pixels fight, then the surviving pixel will be red. As a result, after the fight there are two red pixels. There won't be other pixels. In the second sample the following sequence of fights is possible: red and blue, green and red, red and blue. As a result, after all fights there is one green pixel left. 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. Some days ago, I learned the concept of LCM (least common multiple). I've played with it for several times and I want to make a big number with it. But I also don't want to use many numbers, so I'll choose three positive integers (they don't have to be distinct) which are not greater than n. Can you help me to find the maximum possible least common multiple of these three integers? Input The first line contains an integer n (1 ≤ n ≤ 106) — the n mentioned in the statement. Output Print a single integer — the maximum possible LCM of three not necessarily distinct positive integers that are not greater than n. Examples Input 9 Output 504 Input 7 Output 210 Note The least common multiple of some positive integers is the least positive integer which is multiple for each of them. The result may become very large, 32-bit integer won't be enough. So using 64-bit integers is recommended. For the last example, we can chose numbers 7, 6, 5 and the LCM of them is 7·6·5 = 210. It is the maximum value we can get. 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 Government decided to improve relations with neighboring countries. First of all, it was decided to build new roads so that from each city of Berland and neighboring countries it became possible to reach all the others. There are n cities in Berland and neighboring countries in total and exactly n - 1 two-way roads. Because of the recent financial crisis, the Berland Government is strongly pressed for money, so to build a new road it has to close some of the existing ones. Every day it is possible to close one existing road and immediately build a new one. Your task is to determine how many days would be needed to rebuild roads so that from each city it became possible to reach all the others, and to draw a plan of closure of old roads and building of new ones. Input The first line contains integer n (2 ≤ n ≤ 1000) — amount of cities in Berland and neighboring countries. Next n - 1 lines contain the description of roads. Each road is described by two space-separated integers ai, bi (1 ≤ ai, bi ≤ n, ai ≠ bi) — pair of cities, which the road connects. It can't be more than one road between a pair of cities. No road connects the city with itself. Output Output the answer, number t — what is the least amount of days needed to rebuild roads so that from each city it became possible to reach all the others. Then output t lines — the plan of closure of old roads and building of new ones. Each line should describe one day in the format i j u v — it means that road between cities i and j became closed and a new road between cities u and v is built. Cities are numbered from 1. If the answer is not unique, output any. Examples Input 2 1 2 Output 0 Input 7 1 2 2 3 3 1 4 5 5 6 6 7 Output 1 3 1 3 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. Polycarpus is sure that his life fits the description: "first there is a white stripe, then a black one, then a white one again". So, Polycarpus is sure that this rule is going to fulfill during the next n days. Polycarpus knows that he is in for w good events and b not-so-good events. At least one event is going to take place during each day. As each day is unequivocally characterizes as a part of a white or a black stripe, then each day is going to have events of the same type only (ether good or not-so-good). What is the number of distinct ways this scenario can develop over the next n days if Polycarpus is in for a white stripe (a stripe that has good events only, the stripe's length is at least 1 day), the a black stripe (a stripe that has not-so-good events only, the stripe's length is at least 1 day) and a white stripe again (a stripe that has good events only, the stripe's length is at least 1 day). Each of n days will belong to one of the three stripes only. Note that even the events of the same type are distinct from each other. Even if some events occur on the same day, they go in some order (there are no simultaneous events). Write a code that prints the number of possible configurations to sort the events into days. See the samples for clarifications on which scenarios should be considered distinct. Print the answer modulo 1000000009 (109 + 9). Input The single line of the input contains integers n, w and b (3 ≤ n ≤ 4000, 2 ≤ w ≤ 4000, 1 ≤ b ≤ 4000) — the number of days, the number of good events and the number of not-so-good events. It is guaranteed that w + b ≥ n. Output Print the required number of ways modulo 1000000009 (109 + 9). Examples Input 3 2 1 Output 2 Input 4 2 2 Output 4 Input 3 2 2 Output 4 Note We'll represent the good events by numbers starting from 1 and the not-so-good events — by letters starting from 'a'. Vertical lines separate days. In the first sample the possible ways are: "1\|a\|2" and "2\|a\|1". In the second sample the possible ways are: "1\|a\|b\|2", "2\|a\|b\|1", "1\|b\|a\|2" and "2\|b\|a\|1". In the third sample the possible ways are: "1\|ab\|2", "2\|ab\|1", "1\|ba\|2" and "2\|ba\|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. Pavel loves grid mazes. A grid maze is an n × m rectangle maze where each cell is either empty, or is a wall. You can go from one cell to another only if both cells are empty and have a common side. Pavel drew a grid maze with all empty cells forming a connected area. That is, you can go from any empty cell to any other one. Pavel doesn't like it when his maze has too little walls. He wants to turn exactly k empty cells into walls so that all the remaining cells still formed a connected area. Help him. Input The first line contains three integers n, m, k (1 ≤ n, m ≤ 500, 0 ≤ k < s), where n and m are the maze's height and width, correspondingly, k is the number of walls Pavel wants to add and letter s represents the number of empty cells in the original maze. Each of the next n lines contains m characters. They describe the original maze. If a character on a line equals ".", then the corresponding cell is empty and if the character equals "#", then the cell is a wall. Output Print n lines containing m characters each: the new maze that fits Pavel's requirements. Mark the empty cells that you transformed into walls as "X", the other cells must be left without changes (that is, "." and "#"). It is guaranteed that a solution exists. If there are multiple solutions you can output any of them. Examples Input 3 4 2 #..# ..#. #... Output #.X# X.#. #... Input 5 4 5 #... #.#. .#.. ...# .#.# Output #XXX #X#. X#.. ...# .#.# 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 all know the Dirichlet principle, the point of which is that if n boxes have no less than n + 1 items, that leads to the existence of a box in which there are at least two items. Having heard of that principle, but having not mastered the technique of logical thinking, 8 year olds Stas and Masha invented a game. There are a different boxes and b different items, and each turn a player can either add a new box or a new item. The player, after whose turn the number of ways of putting b items into a boxes becomes no less then a certain given number n, loses. All the boxes and items are considered to be different. Boxes may remain empty. Who loses if both players play optimally and Stas's turn is first? Input The only input line has three integers a, b, n (1 ≤ a ≤ 10000, 1 ≤ b ≤ 30, 2 ≤ n ≤ 109) — the initial number of the boxes, the number of the items and the number which constrains the number of ways, respectively. Guaranteed that the initial number of ways is strictly less than n. Output Output "Stas" if Masha wins. Output "Masha" if Stas wins. In case of a draw, output "Missing". Examples Input 2 2 10 Output Masha Input 5 5 16808 Output Masha Input 3 1 4 Output Stas Input 1 4 10 Output Missing Note In the second example the initial number of ways is equal to 3125. * If Stas increases the number of boxes, he will lose, as Masha may increase the number of boxes once more during her turn. After that any Stas's move will lead to defeat. * But if Stas increases the number of items, then any Masha's move will be losing. 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. Jzzhu have n non-negative integers a1, a2, ..., an. We will call a sequence of indexes i1, i2, ..., ik (1 ≤ i1 < i2 < ... < ik ≤ n) a group of size k. Jzzhu wonders, how many groups exists such that ai1 & ai2 & ... & aik = 0 (1 ≤ k ≤ n)? Help him and print this number modulo 1000000007 (109 + 7). Operation x & y denotes bitwise AND operation of two numbers. Input The first line contains a single integer n (1 ≤ n ≤ 106). The second line contains n integers a1, a2, ..., an (0 ≤ ai ≤ 106). Output Output a single integer representing the number of required groups modulo 1000000007 (109 + 7). Examples Input 3 2 3 3 Output 0 Input 4 0 1 2 3 Output 10 Input 6 5 2 0 5 2 1 Output 53 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 bracket sequences Arthur took up number theory. He has got a new favorite sequence of length n (a1, a2, ..., an), consisting of integers and integer k, not exceeding n. This sequence had the following property: if you write out the sums of all its segments consisting of k consecutive elements (a1 + a2 ... + ak, a2 + a3 + ... + ak + 1, ..., an - k + 1 + an - k + 2 + ... + an), then those numbers will form strictly increasing sequence. For example, for the following sample: n = 5, k = 3, a = (1, 2, 4, 5, 6) the sequence of numbers will look as follows: (1 + 2 + 4, 2 + 4 + 5, 4 + 5 + 6) = (7, 11, 15), that means that sequence a meets the described property. Obviously the sequence of sums will have n - k + 1 elements. Somebody (we won't say who) replaced some numbers in Arthur's sequence by question marks (if this number is replaced, it is replaced by exactly one question mark). We need to restore the sequence so that it meets the required property and also minimize the sum \|ai\|, where \|ai\| is the absolute value of ai. Input The first line contains two integers n and k (1 ≤ k ≤ n ≤ 105), showing how many numbers are in Arthur's sequence and the lengths of segments respectively. The next line contains n space-separated elements ai (1 ≤ i ≤ n). If ai = ?, then the i-th element of Arthur's sequence was replaced by a question mark. Otherwise, ai ( - 109 ≤ ai ≤ 109) is the i-th element of Arthur's sequence. Output If Arthur is wrong at some point and there is no sequence that could fit the given information, print a single string "Incorrect sequence" (without the quotes). Otherwise, print n integers — Arthur's favorite sequence. If there are multiple such sequences, print the sequence with the minimum sum \|ai\|, where \|ai\| is the absolute value of ai. If there are still several such sequences, you are allowed to print any of them. Print the elements of the sequence without leading zeroes. Examples Input 3 2 ? 1 2 Output 0 1 2 Input 5 1 -10 -9 ? -7 -6 Output -10 -9 -8 -7 -6 Input 5 3 4 6 7 2 9 Output Incorrect sequence 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've got array A, consisting of n integers and a positive integer k. Array A is indexed by integers from 1 to n. You need to permute the array elements so that value <image> became minimal possible. In particular, it is allowed not to change order of elements at all. Input The first line contains two integers n, k (2 ≤ n ≤ 3·105, 1 ≤ k ≤ min(5000, n - 1)). The second line contains n integers A[1], A[2], ..., A[n] ( - 109 ≤ A[i] ≤ 109), separate by spaces — elements of the array A. Output Print the minimum possible value of the sum described in the statement. Examples Input 3 2 1 2 4 Output 1 Input 5 2 3 -5 3 -5 3 Output 0 Input 6 3 4 3 4 3 2 5 Output 3 Note In the first test one of the optimal permutations is 1 4 2. In the second test the initial order is optimal. In the third test one of the optimal permutations is 2 3 4 4 3 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. Peter got a new snow blower as a New Year present. Of course, Peter decided to try it immediately. After reading the instructions he realized that it does not work like regular snow blowing machines. In order to make it work, you need to tie it to some point that it does not cover, and then switch it on. As a result it will go along a circle around this point and will remove all the snow from its path. Formally, we assume that Peter's machine is a polygon on a plane. Then, after the machine is switched on, it will make a circle around the point to which Peter tied it (this point lies strictly outside the polygon). That is, each of the points lying within or on the border of the polygon will move along the circular trajectory, with the center of the circle at the point to which Peter tied his machine. Peter decided to tie his car to point P and now he is wondering what is the area of the region that will be cleared from snow. Help him. Input The first line of the input contains three integers — the number of vertices of the polygon n (<image>), and coordinates of point P. Each of the next n lines contains two integers — coordinates of the vertices of the polygon in the clockwise or counterclockwise order. It is guaranteed that no three consecutive vertices lie on a common straight line. All the numbers in the input are integers that do not exceed 1 000 000 in their absolute value. Output Print a single real value number — the area of the region that will be cleared. 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 3 0 0 0 1 -1 2 1 2 Output 12.566370614359172464 Input 4 1 -1 0 0 1 2 2 0 1 1 Output 21.991148575128551812 Note In the first sample snow will be removed from that area: <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, Pari and Arya did some research about NP-Hard problems and they found the minimum vertex cover problem very interesting. Suppose the graph G is given. Subset A of its vertices is called a vertex cover of this graph, if for each edge uv there is at least one endpoint of it in this set, i.e. <image> or <image> (or both). Pari and Arya have won a great undirected graph as an award in a team contest. Now they have to split it in two parts, but both of them want their parts of the graph to be a vertex cover. They have agreed to give you their graph and you need to find two disjoint subsets of its vertices A and B, such that both A and B are vertex cover or claim it's impossible. Each vertex should be given to no more than one of the friends (or you can even keep it for yourself). Input The first line of the input contains two integers n and m (2 ≤ n ≤ 100 000, 1 ≤ m ≤ 100 000) — the number of vertices and the number of edges in the prize graph, respectively. Each of the next m lines contains a pair of integers ui and vi (1 ≤ ui, vi ≤ n), denoting an undirected edge between ui and vi. It's guaranteed the graph won't contain any self-loops or multiple edges. Output If it's impossible to split the graph between Pari and Arya as they expect, print "-1" (without quotes). If there are two disjoint sets of vertices, such that both sets are vertex cover, print their descriptions. Each description must contain two lines. The first line contains a single integer k denoting the number of vertices in that vertex cover, and the second line contains k integers — the indices of vertices. Note that because of m ≥ 1, vertex cover cannot be empty. Examples Input 4 2 1 2 2 3 Output 1 2 2 1 3 Input 3 3 1 2 2 3 1 3 Output -1 Note In the first sample, you can give the vertex number 2 to Arya and vertices numbered 1 and 3 to Pari and keep vertex number 4 for yourself (or give it someone, if you wish). In the second sample, there is no way to satisfy both Pari and Arya. 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. Polycarp starts his own business. Tomorrow will be the first working day of his car repair shop. For now the car repair shop is very small and only one car can be repaired at a given time. Polycarp is good at marketing, so he has already collected n requests from clients. The requests are numbered from 1 to n in order they came. The i-th request is characterized by two values: si — the day when a client wants to start the repair of his car, di — duration (in days) to repair the car. The days are enumerated from 1, the first day is tomorrow, the second day is the day after tomorrow and so on. Polycarp is making schedule by processing requests in the order from the first to the n-th request. He schedules the i-th request as follows: * If the car repair shop is idle for di days starting from si (si, si + 1, ..., si + di - 1), then these days are used to repair a car of the i-th client. * Otherwise, Polycarp finds the first day x (from 1 and further) that there are di subsequent days when no repair is scheduled starting from x. In other words he chooses the smallest positive x that all days x, x + 1, ..., x + di - 1 are not scheduled for repair of any car. So, the car of the i-th client will be repaired in the range [x, x + di - 1]. It is possible that the day x when repair is scheduled to start will be less than si. Given n requests, you are asked to help Polycarp schedule all of them according to the rules above. Input The first line contains integer n (1 ≤ n ≤ 200) — the number of requests from clients. The following n lines contain requests, one request per line. The i-th request is given as the pair of integers si, di (1 ≤ si ≤ 109, 1 ≤ di ≤ 5·106), where si is the preferred time to start repairing the i-th car, di is the number of days to repair the i-th car. The requests should be processed in the order they are given in the input. Output Print n lines. The i-th line should contain two integers — the start day to repair the i-th car and the finish day to repair the i-th car. Examples Input 3 9 2 7 3 2 4 Output 9 10 1 3 4 7 Input 4 1000000000 1000000 1000000000 1000000 100000000 1000000 1000000000 1000000 Output 1000000000 1000999999 1 1000000 100000000 100999999 1000001 2000000 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. Ilya is an experienced player in tic-tac-toe on the 4 × 4 field. He always starts and plays with Xs. He played a lot of games today with his friend Arseny. The friends became tired and didn't finish the last game. It was Ilya's turn in the game when they left it. Determine whether Ilya could have won the game by making single turn or not. The rules of tic-tac-toe on the 4 × 4 field are as follows. Before the first turn all the field cells are empty. The two players take turns placing their signs into empty cells (the first player places Xs, the second player places Os). The player who places Xs goes first, the another one goes second. The winner is the player who first gets three of his signs in a row next to each other (horizontal, vertical or diagonal). Input The tic-tac-toe position is given in four lines. Each of these lines contains four characters. Each character is '.' (empty cell), 'x' (lowercase English letter x), or 'o' (lowercase English letter o). It is guaranteed that the position is reachable playing tic-tac-toe, and it is Ilya's turn now (in particular, it means that the game is not finished). It is possible that all the cells are empty, it means that the friends left without making single turn. Output Print single line: "YES" in case Ilya could have won by making single turn, and "NO" otherwise. Examples Input xx.. .oo. x... oox. Output YES Input x.ox ox.. x.o. oo.x Output NO Input x..x ..oo o... x.xo Output YES Input o.x. o... .x.. ooxx Output NO Note In the first example Ilya had two winning moves: to the empty cell in the left column and to the leftmost empty cell in the first row. In the second example it wasn't possible to win by making single turn. In the third example Ilya could have won by placing X in the last row between two existing Xs. In the fourth example it wasn't possible to win by making single turn. 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. Stepan likes to repeat vowel letters when he writes words. For example, instead of the word "pobeda" he can write "pobeeeedaaaaa". Sergey does not like such behavior, so he wants to write a program to format the words written by Stepan. This program must combine all consecutive equal vowels to a single vowel. The vowel letters are "a", "e", "i", "o", "u" and "y". There are exceptions: if letters "e" or "o" repeat in a row exactly 2 times, like in words "feet" and "foot", the program must skip them and do not transform in one vowel. For example, the word "iiiimpleeemeentatiioon" must be converted to the word "implemeentatioon". Sergey is very busy and asks you to help him and write the required program. Input The first line contains the integer n (1 ≤ n ≤ 100 000) — the number of letters in the word written by Stepan. The second line contains the string s which has length that equals to n and contains only lowercase English letters — the word written by Stepan. Output Print the single string — the word written by Stepan converted according to the rules described in the statement. Examples Input 13 pobeeeedaaaaa Output pobeda Input 22 iiiimpleeemeentatiioon Output implemeentatioon Input 18 aeiouyaaeeiioouuyy Output aeiouyaeeioouy Input 24 aaaoooiiiuuuyyyeeeggghhh Output aoiuyeggghhh 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. Mister B once received a gift: it was a book about aliens, which he started read immediately. This book had c pages. At first day Mister B read v0 pages, but after that he started to speed up. Every day, starting from the second, he read a pages more than on the previous day (at first day he read v0 pages, at second — v0 + a pages, at third — v0 + 2a pages, and so on). But Mister B is just a human, so he physically wasn't able to read more than v1 pages per day. Also, to refresh his memory, every day, starting from the second, Mister B had to reread last l pages he read on the previous day. Mister B finished the book when he read the last page for the first time. Help Mister B to calculate how many days he needed to finish the book. Input First and only line contains five space-separated integers: c, v0, v1, a and l (1 ≤ c ≤ 1000, 0 ≤ l < v0 ≤ v1 ≤ 1000, 0 ≤ a ≤ 1000) — the length of the book in pages, the initial reading speed, the maximum reading speed, the acceleration in reading speed and the number of pages for rereading. Output Print one integer — the number of days Mister B needed to finish the book. Examples Input 5 5 10 5 4 Output 1 Input 12 4 12 4 1 Output 3 Input 15 1 100 0 0 Output 15 Note In the first sample test the book contains 5 pages, so Mister B read it right at the first day. In the second sample test at first day Mister B read pages number 1 - 4, at second day — 4 - 11, at third day — 11 - 12 and finished the book. In third sample test every day Mister B read 1 page of the book, so he finished in 15 days. 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. Alex, Bob and Carl will soon participate in a team chess tournament. Since they are all in the same team, they have decided to practise really hard before the tournament. But it's a bit difficult for them because chess is a game for two players, not three. So they play with each other according to following rules: * Alex and Bob play the first game, and Carl is spectating; * When the game ends, the one who lost the game becomes the spectator in the next game, and the one who was spectating plays against the winner. Alex, Bob and Carl play in such a way that there are no draws. Today they have played n games, and for each of these games they remember who was the winner. They decided to make up a log of games describing who won each game. But now they doubt if the information in the log is correct, and they want to know if the situation described in the log they made up was possible (that is, no game is won by someone who is spectating if Alex, Bob and Carl play according to the rules). Help them to check it! Input The first line contains one integer n (1 ≤ n ≤ 100) — the number of games Alex, Bob and Carl played. Then n lines follow, describing the game log. i-th line contains one integer ai (1 ≤ ai ≤ 3) which is equal to 1 if Alex won i-th game, to 2 if Bob won i-th game and 3 if Carl won i-th game. Output Print YES if the situation described in the log was possible. Otherwise print NO. Examples Input 3 1 1 2 Output YES Input 2 1 2 Output NO Note In the first example the possible situation is: 1. Alex wins, Carl starts playing instead of Bob; 2. Alex wins, Bob replaces Carl; 3. Bob wins. The situation in the second example is impossible because Bob loses the first game, so he cannot win the second 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. Petya and Vasya arranged a game. The game runs by the following rules. Players have a directed graph consisting of n vertices and m edges. One of the vertices contains a chip. Initially the chip is located at vertex s. Players take turns moving the chip along some edge of the graph. Petya goes first. Player who can't move the chip loses. If the game lasts for 106 turns the draw is announced. Vasya was performing big laboratory work in "Spelling and parts of speech" at night before the game, so he fell asleep at the very beginning of the game. Petya decided to take the advantage of this situation and make both Petya's and Vasya's moves. Your task is to help Petya find out if he can win the game or at least draw a tie. Input The first line of input contain two integers n and m — the number of vertices and the number of edges in the graph (2 ≤ n ≤ 105, 0 ≤ m ≤ 2·105). The next n lines contain the information about edges of the graph. i-th line (1 ≤ i ≤ n) contains nonnegative integer ci — number of vertices such that there is an edge from i to these vertices and ci distinct integers ai, j — indices of these vertices (1 ≤ ai, j ≤ n, ai, j ≠ i). It is guaranteed that the total sum of ci equals to m. The next line contains index of vertex s — the initial position of the chip (1 ≤ s ≤ n). Output If Petya can win print «Win» in the first line. In the next line print numbers v1, v2, ..., vk (1 ≤ k ≤ 106) — the sequence of vertices Petya should visit for the winning. Vertex v1 should coincide with s. For i = 1... k - 1 there should be an edge from vi to vi + 1 in the graph. There must be no possible move from vertex vk. The sequence should be such that Petya wins the game. If Petya can't win but can draw a tie, print «Draw» in the only line. Otherwise print «Lose». Examples Input 5 6 2 2 3 2 4 5 1 4 1 5 0 1 Output Win 1 2 4 5 Input 3 2 1 3 1 1 0 2 Output Lose Input 2 2 1 2 1 1 1 Output Draw Note In the first example the graph is the following: <image> Initially the chip is located at vertex 1. In the first move Petya moves the chip to vertex 2, after that he moves it to vertex 4 for Vasya. After that he moves to vertex 5. Now it is Vasya's turn and there is no possible move, so Petya wins. In the second example the graph is the following: <image> Initially the chip is located at vertex 2. The only possible Petya's move is to go to vertex 1. After that he has to go to 3 for Vasya. Now it's Petya's turn but he has no possible move, so Petya loses. In the third example the graph is the following: <image> Petya can't win, but he can move along the cycle, so the players will draw a tie. 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. Roy is looking for Wobbly Numbers. An N-length wobbly number is of the form "ababababab..." and so on of length N, where a != b. A 3-length wobbly number would be of form "aba". Eg: 101, 121, 131, 252, 646 etc But 111, 222, 999 etc are not 3-length wobbly number, because here a != b condition is not satisfied. Also 010 is not a 3-length wobbly number because it has preceding 0. So 010 equals 10 and 10 is not a 3-length wobbly number. A 4-length wobbly number would be of form "abab". Eg: 2323, 3232, 9090, 1414 etc Similarly we can form a list of N-length wobbly numbers. Now your task is to find K^th wobbly number from a lexicographically sorted list of N-length wobbly numbers. If the number does not exist print -1 else print the K^th wobbly number. See the sample test case and explanation for more clarity. Input: First line contains T - number of test cases Each of the next T lines contains two space separated integers - N and K. Output: For each test case print the required output in a new line. Constraints: 1 ≤ T ≤ 100 3 ≤ N ≤ 1000 1 ≤ K ≤ 100 SAMPLE INPUT 6 3 1 3 2 3 100 4 3 4 4 5 2 SAMPLE OUTPUT 101 121 -1 1313 1414 12121 Explanation First 10 terms of 3-length wobbly numbers arranged lexicographically is as follows: 101, 121, 131, 141, 151, 161, 171, 181, 191, 202 1st wobbly number of length 3 is 101. 2nd wobbly number of length 3 is 121. 100th wobbly number of length 3 does not exist, so the output is -1. First 10 terms of 4-length wobbly numbers arranged lexicographically is as follows: 1010, 1212, 1313, 1414, 1515, 1616, 1717, 1818, 1919, 2020 3rd wobbly number of length 4 is 1313. 4th wobbly number of length 4 is 1414. Similarly 2nd wobbly number of length 5 is 12121 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 code world all genders are considered equal ( It means their is nothing like male or female). Now their are N distinct persons living in this hypothetical world. Each person can pair up with any other person or can even remain single. One day Vbhu planned to visit code world. Being a maths guy , he always try to be mathematical. So he started counting the ways in which N persons living in code world can make pairs or remain single. A single person can make pair with at most one other person.Seeing that N can be large , Vibhu ask you for help. Now being a great programmer you need to help Vbhu count the number of ways in which N persons living in code world can make pairs or remain single. Note : Its not necessary that everyone is required to make pair with someone. Person can remain single also. Input Format : First line contain number of test cases T. Then next T lines contain a single integer N , denoting the number of persons living in code world. Output Format : You need to print the number of ways in which N different persons can make their pairs or stay single. As answer can be large so print it modulo 10^9+7. Constraints : 1 ≤ T ≤10^5 1 ≤ N ≤10^6 Warning: Large Input/Output data, be careful with certain languages SAMPLE INPUT 2 2 3 SAMPLE OUTPUT 2 4 Explanation In first test case , For N=2 answer will be 2. Possible ways are : {1},{2} (It means Person 1 and Person 2 are single) {1,2} (It means Person 1 and Person 2 had formed a pair) For second test case , For N=3 , answer will be 4. Possible ways are : {1},{2},{3} (It means all three Persons are single) {1,2},{3} (It means Person 1 and Person 2 had formed a pair and Person 3 is single) {1},{2,3} (It means Person 2 and Person 3 had formed a pair and Person 1 is single) {1,3},{2} (It means Person 1 and Person 3 had formed a pair and Person 2 is single) 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. Print the K-th element of the following sequence of length 32: 1, 1, 1, 2, 1, 2, 1, 5, 2, 2, 1, 5, 1, 2, 1, 14, 1, 5, 1, 5, 2, 2, 1, 15, 2, 2, 5, 4, 1, 4, 1, 51 Constraints * 1 \leq K \leq 32 * All values in input are integers. Input Input is given from Standard Input in the following format: K Output Print the K-th element. Examples Input 6 Output 2 Input 27 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. We have N balance beams numbered 1 to N. The length of each beam is 1 meters. Snuke walks on Beam i at a speed of 1/A_i meters per second, and Ringo walks on Beam i at a speed of 1/B_i meters per second. Snuke and Ringo will play the following game: * First, Snuke connects the N beams in any order of his choice and makes a long beam of length N meters. * Then, Snuke starts at the left end of the long beam. At the same time, Ringo starts at a point chosen uniformly at random on the long beam. Both of them walk to the right end of the long beam. * Snuke wins if and only if he catches up to Ringo before Ringo reaches the right end of the long beam. That is, Snuke wins if there is a moment when Snuke and Ringo stand at the same position, and Ringo wins otherwise. Find the probability that Snuke wins when Snuke arranges the N beams so that the probability of his winning is maximized. This probability is a rational number, so we ask you to represent it as an irreducible fraction P/Q (to represent 0, use P=0, Q=1). Constraints * 1 \leq N \leq 10^5 * 1 \leq A_i,B_i \leq 10^9 * All values in input are integers. Input Input is given from Standard Input in the following format: N A_1 B_1 A_2 B_2 \vdots A_N B_N Output Print the numerator and denominator of the irreducible fraction that represents the maximum probability of Snuke's winning. Examples Input 2 3 2 1 2 Output 1 4 Input 4 1 5 4 7 2 1 8 4 Output 1 2 Input 3 4 1 5 2 6 3 Output 0 1 Input 10 866111664 178537096 705445072 318106937 472381277 579910117 353498483 865935868 383133839 231371336 378371075 681212831 304570952 16537461 955719384 267238505 844917655 218662351 550309930 62731178 Output 697461712 2899550585 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. Diverta City is a new city consisting of N towns numbered 1, 2, ..., N. The mayor Ringo is planning to connect every pair of two different towns with a bidirectional road. The length of each road is undecided. A Hamiltonian path is a path that starts at one of the towns and visits each of the other towns exactly once. The reversal of a Hamiltonian path is considered the same as the original Hamiltonian path. There are N! / 2 Hamiltonian paths. Ringo wants all these paths to have distinct total lengths (the sum of the lengths of the roads on a path), to make the city diverse. Find one such set of the lengths of the roads, under the following conditions: * The length of each road must be a positive integer. * The maximum total length of a Hamiltonian path must be at most 10^{11}. Constraints * N is a integer between 2 and 10 (inclusive). Input Input is given from Standard Input in the following format: N Output Print a set of the lengths of the roads that meets the objective, in the following format: w_{1, 1} \ w_{1, 2} \ w_{1, 3} \ ... \ w_{1, N} w_{2, 1} \ w_{2, 2} \ w_{2, 3} \ ... \ w_{2, N} : : : w_{N, 1} \ w_{N, 2} \ w_{N, 3} \ ... \ w_{N, N} where w_{i, j} is the length of the road connecting Town i and Town j, which must satisfy the following conditions: * w_{i, i} = 0 * w_{i, j} = w_{j, i} \ (i \neq j) * 1 \leq w_{i, j} \leq 10^{11} \ (i \neq j) If there are multiple sets of lengths of the roads that meet the objective, any of them will be accepted. Examples Input 3 Output 0 6 15 6 0 21 15 21 0 Input 4 Output 0 111 157 193 111 0 224 239 157 224 0 258 193 239 258 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. A string is called a KEYENCE string when it can be changed to `keyence` by removing its contiguous substring (possibly empty) only once. Given a string S consisting of lowercase English letters, determine if S is a KEYENCE string. Constraints * The length of S is between 7 and 100 (inclusive). * S consists of lowercase English letters. Input Input is given from Standard Input in the following format: S Output If S is a KEYENCE string, print `YES`; otherwise, print `NO`. Examples Input keyofscience Output YES Input mpyszsbznf Output NO Input ashlfyha Output NO Input keyence 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. You are given N positive integers a_1, a_2, ..., a_N. For a non-negative integer m, let f(m) = (m\ mod\ a_1) + (m\ mod\ a_2) + ... + (m\ mod\ a_N). Here, X\ mod\ Y denotes the remainder of the division of X by Y. Find the maximum value of f. Constraints * All values in input are integers. * 2 \leq N \leq 3000 * 2 \leq a_i \leq 10^5 Input Input is given from Standard Input in the following format: N a_1 a_2 ... a_N Output Print the maximum value of f. Examples Input 3 3 4 6 Output 10 Input 5 7 46 11 20 11 Output 90 Input 7 994 518 941 851 647 2 581 Output 4527 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 permutation p_1,p_2,...,p_N consisting of 1,2,..,N. You can perform the following operation any number of times (possibly zero): Operation: Swap two adjacent elements in the permutation. You want to have p_i ≠ i for all 1≤i≤N. Find the minimum required number of operations to achieve this. Constraints * 2≤N≤10^5 * p_1,p_2,..,p_N is a permutation of 1,2,..,N. Input The input is given from Standard Input in the following format: N p_1 p_2 .. p_N Output Print the minimum required number of operations Examples Input 5 1 4 3 5 2 Output 2 Input 2 1 2 Output 1 Input 2 2 1 Output 0 Input 9 1 2 4 9 5 8 7 3 6 Output 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. Imagine a game played on a line. Initially, the player is located at position 0 with N candies in his possession, and the exit is at position E. There are also N bears in the game. The i-th bear is located at x_i. The maximum moving speed of the player is 1 while the bears do not move at all. When the player gives a candy to a bear, it will provide a coin after T units of time. More specifically, if the i-th bear is given a candy at time t, it will put a coin at its position at time t+T. The purpose of this game is to give candies to all the bears, pick up all the coins, and go to the exit. Note that the player can only give a candy to a bear if the player is at the exact same position of the bear. Also, each bear will only produce a coin once. If the player visits the position of a coin after or at the exact same time that the coin is put down, the player can pick up the coin. Coins do not disappear until collected by the player. Shik is an expert of this game. He can give candies to bears and pick up coins instantly. You are given the configuration of the game. Please calculate the minimum time Shik needs to collect all the coins and go to the exit. Constraints * 1 \leq N \leq 100,000 * 1 \leq T, E \leq 10^9 * 0 < x_i < E * x_i < x_{i+1} for 1 \leq i < N * All input values are integers. Input The input is given from Standard Input in the following format: N E T x_1 x_2 ... x_N Output Print an integer denoting the answer. Examples Input 3 9 1 1 3 8 Output 12 Input 3 9 3 1 3 8 Output 16 Input 2 1000000000 1000000000 1 999999999 Output 2999999996 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 20XX, the Aizu Chuo Road, which has a total distance of 58km and 6 sections from Atsushiokanomachi, Kitakata City to Minamiaizucho, is scheduled to be completed and opened. For half a year after opening, the toll will be halved for vehicles that pass the departure IC or arrival IC between 17:30 and 19:30 and have a mileage of 40km or less. However, the charge will be in units of 50 yen and rounded up. The table below is a list of fares and distances. <image> <image> For example, from Kitakata (2) to Aizuwakamatsu (4), the fare is 450 yen and the distance is 12km. If it is half price time zone, it will be 250 yen. Create a program that calculates and outputs the charge by inputting the departure IC, departure IC transit time, arrival IC, and arrival IC transit time. However, the time entered will be the value in 24-hour notation. In addition, even if you pass at exactly 17:30 and 19:30, it will be included in the half price time zone. 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: d hd md a ha ma The first line gives the departure IC number d (1 ≤ d ≤ 7), and the second line gives the time hd (0 ≤ hd ≤ 23) and minutes md (0 ≤ md ≤ 59) of the departure IC transit time. .. The arrival IC number a (1 ≤ a ≤ 7) is given on the third line, and the time ha (0 ≤ ha ≤ 23) and minute ma (0 ≤ ma ≤ 59) of the arrival IC transit time are given on the fourth line. .. Output The toll (integer) is output to one line for each data set. Example Input 2 17 25 4 17 45 4 17 25 7 19 35 0 Output 250 1300 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. problem Chairman K is a regular customer of the JOI pizza shop in the center of JOI city. For some reason, he decided to start a life-saving life this month. So he wanted to order the pizza with the highest calories per dollar among the pizzas he could order at the JOI pizza store. Let's call such a pizza the "best pizza". The "best pizza" is not limited to one type. At JOI Pizza, you can freely choose from N types of toppings and order the ones placed on the basic dough. You cannot put more than one topping of the same type. You can also order a pizza that doesn't have any toppings on the dough. The price of the dough is $ A and the price of the toppings is $ B. The price of pizza is the sum of the price of the dough and the price of the toppings. That is, the price of a pizza with k types of toppings (0 ≤ k ≤ N) is A + k x B dollars. The total calorie of the pizza is the sum of the calories of the dough and the calories of the toppings placed. Create a program to find the number of calories per dollar for the "best pizza" given the price of the dough and the price of the toppings, and the calorie value of the dough and each topping. input The input consists of N + 3 lines. On the first line, one integer N (1 ≤ N ≤ 100) representing the number of topping types is written. On the second line, two integers A and B (1 ≤ A ≤ 1000, 1 ≤ B ≤ 1000) are written with a blank as a delimiter. A is the price of the dough and B is the price of the toppings. On the third line, one integer C (1 ≤ C ≤ 10000) representing the number of calories in the dough is written. On the 3 + i line (1 ≤ i ≤ N), one integer Di (1 ≤ Di ≤ 10000) representing the number of calories in the i-th topping is written. output Print the number of calories per dollar for the "best pizza" in one line. However, round down the numbers after the decimal point and output as an integer value. Input / output example Input example 1 3 12 2 200 50 300 100 Output example 1 37 In I / O Example 1, with the second and third toppings, 200 + 300 + 100 = 600 calories gives a pizza of $ 12 + 2 x 2 = $ 16. This pizza has 600/16 = 37.5 calories per dollar. Since this is the "best pizza", we output 37, rounded down to the nearest whole number of 37.5. Input example 2 Four 20 3 900 300 100 400 1300 Output example 2 100 The question text and the data used for the automatic referee are the question text and the test data for scoring, which are created and published by the Japan Committee for Information Olympics. Example Input 3 12 2 200 50 300 100 Output 37 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. Problem KND is a student programmer at the University of Aizu. His chest is known to be very sexy. <image> For simplicity, the part of the skin that can be seen from the chest is represented by the isosceles triangle ABC in the figure. However, due to the slack in the clothes, the two sides AC and BC (where these lengths are l), which have the same length, actually have an additional length x minutes. In order to increase the area of the open part, let's make two new triangular ADCs and BECs by pulling the slack part. Points D and E exist outside the triangle ABC. These two new triangles are caused by slack, and the sum of the lengths of side BE and side EC and the sum of the lengths of side AD and side DC must be l + x. You determine the points D and E so that the sum M of the areas of these three triangles is maximized. As KND's neighbor, you decide to write a program to calculate the maximum area of skin (M) to look out of your clothes, using a, l, x as inputs to find out how sexy his chest is. did. Constraints The input satisfies the following conditions. * All inputs are integers. * 1 ≤ a ≤ 1000 * 1 ≤ l ≤ 1000 * 1 ≤ x ≤ 1000 Input The input consists of multiple test cases. One test case is given in the following format. The end of input is indicated by EOF. a l x here, * a: Length of side AB of triangle ABC * l: Length of two sides AC and BC of triangle ABC * x: Slack on two sides AC, BC Is. Output Output the maximum area for each test case on one line. This value should not differ more than 10-5 from the value of the judge output. Example Input 2 2 1 2 3 1 3 2 3 2 3 5 Output 3.9681187851 6.7970540913 6.5668891783 13.9527248554 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 King of a little Kingdom on a little island in the Pacific Ocean frequently has childish ideas. One day he said, “You shall make use of a message relaying game when you inform me of something.” In response to the King’s statement, six servants were selected as messengers whose names were Mr. J, Miss C, Mr. E, Mr. A, Dr. P, and Mr. M. They had to relay a message to the next messenger until the message got to the King. Messages addressed to the King consist of digits (‘0’-‘9’) and alphabet characters (‘a’-‘z’, ‘A’-‘Z’). Capital and small letters are distinguished in messages. For example, “ke3E9Aa” is a message. Contrary to King’s expectations, he always received wrong messages, because each messenger changed messages a bit before passing them to the next messenger. Since it irritated the King, he told you who are the Minister of the Science and Technology Agency of the Kingdom, “We don’t want such a wrong message any more. You shall develop software to correct it!” In response to the King’s new statement, you analyzed the messengers’ mistakes with all technologies in the Kingdom, and acquired the following features of mistakes of each messenger. A surprising point was that each messenger made the same mistake whenever relaying a message. The following facts were observed. Mr. J rotates all characters of the message to the left by one. For example, he transforms “aB23d” to “B23da”. Miss C rotates all characters of the message to the right by one. For example, she transforms “aB23d” to “daB23”. Mr. E swaps the left half of the message with the right half. If the message has an odd number of characters, the middle one does not move. For example, he transforms “e3ac” to “ace3”, and “aB23d” to “3d2aB”. Mr. A reverses the message. For example, he transforms “aB23d” to “d32Ba”. Dr. P increments by one all the digits in the message. If a digit is ‘9’, it becomes ‘0’. The alphabet characters do not change. For example, he transforms “aB23d” to “aB34d”, and “e9ac” to “e0ac”. Mr. M decrements by one all the digits in the message. If a digit is ‘0’, it becomes ‘9’. The alphabet characters do not change. For example, he transforms “aB23d” to “aB12d”, and “e0ac” to “e9ac”. The software you must develop is to infer the original message from the final message, given the order of the messengers. For example, if the order of the messengers is A -> J -> M -> P and the message given to the King is “aB23d”, what is the original message? According to the features of the messengers’ mistakes, the sequence leading to the final message is A J M P “32Bad” --> “daB23” --> “aB23d” --> “aB12d” --> “aB23d”. As a result, the original message should be “32Bad”. Input The input format is as follows. n The order of messengers The message given to the King . . . The order of messengers The message given to the King The first line of the input contains a positive integer n, which denotes the number of data sets. Each data set is a pair of the order of messengers and the message given to the King. The number of messengers relaying a message is between 1 and 6 inclusive. The same person may not appear more than once in the order of messengers. The length of a message is between 1 and 25 inclusive. Output The inferred messages are printed each on a separate line. Example Input 5 AJMP aB23d E 86AE AM 6 JPEM WaEaETC302Q CP rTurnAGundam1isdefferentf Output 32Bad AE86 7 EC302QTWaEa TurnAGundam0isdefferentfr 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. Example Input ACM 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. A rabbit is playing a role-playing game. Just before entering the castle, he was ambushed by an enemy! It was a battle between one hero operated by a rabbit and n enemies. Each character has four stats, health hi, attack power ai, defense power di, and agility si. I = 0 is the information of the main character, 1 ≤ i ≤ n is the information of each enemy. The battle is turn-based. Each turn, the surviving characters attack in descending order of agility. The enemy always attacks the hero. The hero attacks one enemy, but which enemy to attack Can be selected by the main character every turn. When a character with attack power a attacks a character with defense power d, max {a − d, 0} damage is dealt. The total damage received is greater than or equal to the value of physical strength. The character becomes incapacitated immediately. The battle ends when the main character becomes incapacitated, or when all the enemies become incapacitated. Input 1 ≤ n ≤ 40 000 1 ≤ hi, ai, di, si ≤ 1 000 000 000 (integer) si are all different. Output When the hero is sure to be incapacitated, output -1. Otherwise, output the minimum total damage to the hero in one line. Examples Input 2 10 3 1 2 2 4 1 3 2 2 1 1 Output 4 Input 1 1 1 1 1 10000 10000 10000 10000 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. D: The Diversity of Prime Factorization Problem Ebi-chan has the FACTORIZATION MACHINE, which can factorize natural numbers M (greater than 1) in O ($ \ log $ M) time! But unfortunately, the machine could display only digits and white spaces. In general, we consider the factorization of M as p_1 ^ {e_1} \ times p_2 ^ {e_2} \ times ... \ times p_K ^ {e_K} where (1) i <j implies p_i <p_j and (2) p_i is prime. Now, she gives M to the machine, and the machine displays according to the following rules in ascending order with respect to i: * If e_i = 1, then displays p_i, * otherwise, displays p_i e_i. For example, if she gives either `22` or` 2048`, then `2 11` is displayed. If either` 24` or `54`, then` 2 3 3`. Okay, Ebi-chan has written down the output of the machine, but she notices that she has forgotten to write down the input! Now, your task is to count how many natural numbers result in a noted output. Note that Ebi-chan has mistaken writing and no input could result in the output. The answer could be too large, so, you must output it modulo 10 ^ 9 + 7 (prime number). Input N q_1 q_2 $ \ cdots $ q_N In the first line, the number of the output of the machine is given. In the second line, the output of the machine is given. Constraints * 1 \ leq N \ leq 10 ^ 5 * 2 \ leq q_i \ leq 10 ^ 6 (1 \ leq i \ leq N) Output Print the number of the natural numbers that result in the given output of the machine. Sample Input 1 3 2 3 3 Sample Output for Input 1 2 24 = 2 ^ 3 \ times 3 and 54 = 2 \ times 3 ^ 3 satisfy the condition. Sample Input 2 3 2 3 4 Sample Output 2 for Input 2 1 Only 162 = 2 \ times 3 ^ 4 satisfies the condition. Note that 4 is not prime. Sample Input 3 3 3 5 2 Sample Output for Input 3 1 Since 2 <3 <5, only 75 = 3 \ times 5 ^ 2 satisfies the condition. Sample Input 4 1 Four Sample Output for Input 4 0 Ebi-chan should have written down it more carefully. Example Input 3 2 3 3 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.

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