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Solve the programming task below in a Python markdown code block. Nobody knows, but $N$ frogs live in Chef's garden. Now they are siting on the X-axis and want to speak to each other. One frog can send a message to another one if the distance between them is less or equal to $K$. Chef knows all $P$ pairs of frogs, which want to send messages. Help him to define can they or not! Note : More than $1$ frog can be on the same point on the X-axis. -----Input----- - The first line contains three integers $N$, $K$ and $P$. - The second line contains $N$ space-separated integers $A_1$, $A_2$, …, $A_N$ denoting the x-coordinates of frogs". - Each of the next $P$ lines contains two integers $A$ and $B$ denoting the numbers of frogs according to the input. -----Output----- For each pair print "Yes" without a brackets if frogs can speak and "No" if they cannot. -----Constraints----- - $1 \le N, P \le 10^5$ - $0 \le A_i, K \le 10^9$ - $1 \le A, B \le N$ -----Example----- -----Sample Input:----- 5 3 3 0 3 8 5 12 1 2 1 3 2 5 -----Sample Output:----- Yes Yes No -----Explanation----- - For pair $(1, 2)$ frog $1$ can directly speak to the frog $2$ as the distance between them is $3 - 0 = 3 \le K$ . - For pair $(1, 3)$ frog $1$ can send a message to frog $2$, frog $2$ can send it to frog $4$ and it can send it to frog $3$. - For pair $(2, 5)$ frogs can't send a message under current constraints. 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. [0, 0] [0, 1] [1, 1] [0, 2] [1, 2] [2, 2] [0, 3] [1, 3] [2, 3] [3, 3] [0, 4] [1, 4] [2, 4] [3, 4] [4, 4] [0, 5] [1, 5] [2, 5] [3, 5] [4, 5] [5, 5] [0, 6] [1, 6] [2, 6] [3, 6] [4, 6] [5, 6] [6, 6] Consider the standard set of 28 western dominoes as shown in the above figure. Given a subset of the standard set dominoes, decide whether this subset can be arranged in a straight row in accordance with the familiar playing rule that touching ends must match. For example, the subset [1, 1], [2, 2], [1, 2] can be arranged in a row (as [1, 1] followed by [1, 2] followed by [2, 2]), while the subset [1, 1], [0, 3], [1, 4] can not be arranged in one row. Note that as in usual dominoes playing any pair [i, j] can also be treated as [j, i]. Your task is to write a program that takes as input any subset of the dominoes and output either yes (if the input subset can be arranged in one row) or no (if the input set can not be arranged in one row). Input Input file consists of pairs of lines. The first line of each pair is the number of elements N (1 ≤ N ≤ 18) in the subset, and the second line is the elements of the subset separated by blanks, see the input sample below. The number of pairs (datasets) is less than 30. Output For each pair of lines of the input file, the corresponding output is either Yes or No, based on whether the input subset can be arranged in one line or not. Example Input 6 13 23 14 24 15 25 10 00 01 11 02 12 22 03 13 23 33 Output Yes No 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. Jack loved his house very much, because his lovely cats take a nap on the wall of his house almost every day. Jack loved cats very much. Jack decided to keep an observation diary of the cats as a free study during the summer vacation. After observing for a while, he noticed an interesting feature of the cats. The fence has a width of W [scale], and the cats take a nap side by side. Since each cat has a different body size, the width required for a nap is also different. I came to take a nap. Cats take a nap there if they have a place to sleep. However, if there are multiple such places, they take a nap on the left side, and if there is not enough width, they give up and go home. Take a nap for a while. When the nap cat gets up, it jumps off the wall and goes somewhere. Jack observed the cats and wrote down their behavior in a notebook. However, it is very difficult to compile this record because so many cats were taking a nap. Writing a program, the cats Help me find a place to take a nap. Input The input file contains multiple datasets. The first line of each dataset contains two integers, each representing the width W of the fence and the number of lines Q that follow. The following Q line is given the observation record of the cat. Each line follows one of the following formats. s [id] [w] w [id] The former is a cat's sleep record, which indicates that the cat came to take a nap. Id is an integer that represents the cat's name, and w is the width [scale] that the cat needs to take a nap. The latter is a wakeup record of the cat, indicating that a cat with the name id has occurred. This record is given in chronological order. No cat will take a nap more than once. Also, Jack's record shall be consistent. That is, if a cat can take a nap against a cat's sleep record, then Only if the cat's wakeup record is given (on a later line). You can assume that W, Q ≤ 100. When both W and Q are 0, it indicates the end of input. Do not output to this dataset. Output Output one line each time a sleep record of input is given. This line must contain a number that indicates the position of the cat if it can take a nap. The cat starts at the left edge of the wall and is b [scale. ] To b + w [shaku], output b. If the cat could not take a nap, output "impossible". Print "END" at the end of the dataset. Example Input 4 6 s 0 2 s 1 3 s 2 1 w 0 s 3 3 s 4 2 3 3 s 0 1 s 1 1 s 2 1 0 0 Output 0 impossible 2 impossible 0 END 0 1 2 END 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 squirrel Chokudai has N acorns. One day, he decides to do some trades in multiple precious metal exchanges to make more acorns. His plan is as follows: 1. Get out of the nest with N acorns in his hands. 2. Go to Exchange A and do some trades. 3. Go to Exchange B and do some trades. 4. Go to Exchange A and do some trades. 5. Go back to the nest. In Exchange X (X = A, B), he can perform the following operations any integer number of times (possibly zero) in any order: * Lose g_{X} acorns and gain 1 gram of gold. * Gain g_{X} acorns and lose 1 gram of gold. * Lose s_{X} acorns and gain 1 gram of silver. * Gain s_{X} acorns and lose 1 gram of silver. * Lose b_{X} acorns and gain 1 gram of bronze. * Gain b_{X} acorns and lose 1 gram of bronze. Naturally, he cannot perform an operation that would leave him with a negative amount of acorns, gold, silver, or bronze. What is the maximum number of acorns that he can bring to the nest? Note that gold, silver, or bronze brought to the nest would be worthless because he is just a squirrel. Constraints * 1 \leq N \leq 5000 * 1 \leq g_{X} \leq 5000 * 1 \leq s_{X} \leq 5000 * 1 \leq b_{X} \leq 5000 * All values in input are integers. Input Input is given from Standard Input in the following format: N g_A s_A b_A g_B s_B b_B Output Print the maximum number of acorns that Chokudai can bring to the nest. Example Input 23 1 1 1 2 1 1 Output 46 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 has a simple graph (that is, a graph without loops or multiple edges) consisting of $n$ vertices and $m$ edges. The weight of the $i$-th vertex is $a_i$. The weight of the $i$-th edge is $w_i$. A subgraph of a graph is some set of the graph vertices and some set of the graph edges. The set of edges must meet the condition: both ends of each edge from the set must belong to the chosen set of vertices. The weight of a subgraph is the sum of the weights of its edges, minus the sum of the weights of its vertices. You need to find the maximum weight of subgraph of given graph. The given graph does not contain loops and multiple edges. -----Input----- The first line contains two numbers $n$ and $m$ ($1 \le n \le 10^3, 0 \le m \le 10^3$) - the number of vertices and edges in the graph, respectively. The next line contains $n$ integers $a_1, a_2, \dots, a_n$ ($1 \le a_i \le 10^9$) - the weights of the vertices of the graph. The following $m$ lines contain edges: the $i$-e edge is defined by a triple of integers $v_i, u_i, w_i$ ($1 \le v_i, u_i \le n, 1 \le w_i \le 10^9, v_i \neq u_i$). This triple means that between the vertices $v_i$ and $u_i$ there is an edge of weight $w_i$. It is guaranteed that the graph does not contain loops and multiple edges. -----Output----- Print one integer — the maximum weight of the subgraph of the given graph. -----Examples----- Input 4 5 1 5 2 2 1 3 4 1 4 4 3 4 5 3 2 2 4 2 2 Output 8 Input 3 3 9 7 8 1 2 1 2 3 2 1 3 3 Output 0 -----Note----- In the first test example, the optimal subgraph consists of the vertices ${1, 3, 4}$ and has weight $4 + 4 + 5 - (1 + 2 + 2) = 8$. In the second test case, the optimal subgraph is empty. 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. Read problems statements in Mandarin Chinese and Russian. Tuzik is a little dog. But despite the fact he is still a puppy he already knows about the pretty things that coins are. He knows that for every coin he can get very tasty bone from his master. He believes that some day he will find a treasure and have loads of bones. And finally he found something interesting. A wooden chest containing N coins! But as you should remember, Tuzik is just a little dog, and so he can't open it by himself. Actually, the only thing he can really do is barking. He can use his barking to attract nearby people and seek their help. He can set the loudness of his barking very precisely, and therefore you can assume that he can choose to call any number of people, from a minimum of 1, to a maximum of K. When people come and open the chest they divide all the coins between them in such a way that everyone will get the same amount of coins and this amount is maximal possible. If some coins are not used they will leave it on the ground and Tuzik will take them after they go away. Since Tuzik is clearly not a fool, he understands that his profit depends on the number of people he will call. While Tuzik works on his barking, you have to find the maximum possible number of coins he can get. ------ Input ------ The first line of the input contains an integer T denoting the number of test cases. Each of next T lines contains 2 space-separated integers: N and K, for this test case. ------ Output ------ For each test case output one integer - the maximum possible number of coins Tuzik can get. ------ Constraints ------ $1 ≤ T ≤ 50$ $1 ≤ N, K ≤ 10^{5}$ ----- Sample Input 1 ------ 2 5 2 11 3 ----- Sample Output 1 ------ 1 2 ----- explanation 1 ------ In the first example he should call two people. Each of them will take 2 coins and they will leave 1 coin for Tuzik. In the second example he should call 3 people. 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. Paul hates palindromes. He assumes that string s is tolerable if each its character is one of the first p letters of the English alphabet and s doesn't contain any palindrome contiguous substring of length 2 or more. Paul has found a tolerable string s of length n. Help him find the lexicographically next tolerable string of the same length or else state that such string does not exist. -----Input----- The first line contains two space-separated integers: n and p (1 ≤ n ≤ 1000; 1 ≤ p ≤ 26). The second line contains string s, consisting of n small English letters. It is guaranteed that the string is tolerable (according to the above definition). -----Output----- If the lexicographically next tolerable string of the same length exists, print it. Otherwise, print "NO" (without the quotes). -----Examples----- Input 3 3 cba Output NO Input 3 4 cba Output cbd Input 4 4 abcd Output abda -----Note----- String s is lexicographically larger (or simply larger) than string t with the same length, if there is number i, such that s_1 = t_1, ..., s_{i} = t_{i}, s_{i} + 1 > t_{i} + 1. The lexicographically next tolerable string is the lexicographically minimum tolerable string which is larger than the given one. A palindrome is a string that reads the same forward or reversed. Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. In some other world, today is Christmas. Mr. Takaha decides to make a multi-dimensional burger in his party. A level-L burger (L is an integer greater than or equal to 0) is the following thing: - A level-0 burger is a patty. - A level-L burger (L \geq 1) is a bun, a level-(L-1) burger, a patty, another level-(L-1) burger and another bun, stacked vertically in this order from the bottom. For example, a level-1 burger and a level-2 burger look like BPPPB and BBPPPBPBPPPBB (rotated 90 degrees), where B and P stands for a bun and a patty. The burger Mr. Takaha will make is a level-N burger. Lunlun the Dachshund will eat X layers from the bottom of this burger (a layer is a patty or a bun). How many patties will she eat? -----Constraints----- - 1 \leq N \leq 50 - 1 \leq X \leq ( the total number of layers in a level-N burger ) - N and X are integers. -----Input----- Input is given from Standard Input in the following format: N X -----Output----- Print the number of patties in the bottom-most X layers from the bottom of a level-N burger. -----Sample Input----- 2 7 -----Sample Output----- 4 There are 4 patties in the bottom-most 7 layers of a level-2 burger (BBPPPBPBPPPBB). 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 table consisting of n rows and m columns. Each cell of the table contains a number, 0 or 1. In one move we can choose some row of the table and cyclically shift its values either one cell to the left, or one cell to the right. To cyclically shift a table row one cell to the right means to move the value of each cell, except for the last one, to the right neighboring cell, and to move the value of the last cell to the first cell. A cyclical shift of a row to the left is performed similarly, but in the other direction. For example, if we cyclically shift a row "00110" one cell to the right, we get a row "00011", but if we shift a row "00110" one cell to the left, we get a row "01100". Determine the minimum number of moves needed to make some table column consist only of numbers 1. Input The first line contains two space-separated integers: n (1 ≤ n ≤ 100) — the number of rows in the table and m (1 ≤ m ≤ 104) — the number of columns in the table. Then n lines follow, each of them contains m characters "0" or "1": the j-th character of the i-th line describes the contents of the cell in the i-th row and in the j-th column of the table. It is guaranteed that the description of the table contains no other characters besides "0" and "1". Output Print a single number: the minimum number of moves needed to get only numbers 1 in some column of the table. If this is impossible, print -1. Examples Input 3 6 101010 000100 100000 Output 3 Input 2 3 111 000 Output -1 Note In the first sample one way to achieve the goal with the least number of moves is as follows: cyclically shift the second row to the right once, then shift the third row to the left twice. Then the table column before the last one will contain only 1s. In the second sample one can't shift the rows to get a column containing only 1s. 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. Learn, learn and learn again — Valera has to do this every day. He is studying at mathematical school, where math is the main discipline. The mathematics teacher loves her discipline very much and tries to cultivate this love in children. That's why she always gives her students large and difficult homework. Despite that Valera is one of the best students, he failed to manage with the new homework. That's why he asks for your help. He has the following task. A sequence of n numbers is given. A prefix of a sequence is the part of the sequence (possibly empty), taken from the start of the sequence. A suffix of a sequence is the part of the sequence (possibly empty), taken from the end of the sequence. It is allowed to sequentially make two operations with the sequence. The first operation is to take some prefix of the sequence and multiply all numbers in this prefix by - 1. The second operation is to take some suffix and multiply all numbers in it by - 1. The chosen prefix and suffix may intersect. What is the maximum total sum of the sequence that can be obtained by applying the described operations? Input The first line contains integer n (1 ≤ n ≤ 105) — amount of elements in the sequence. The second line contains n integers ai ( - 104 ≤ ai ≤ 104) — the sequence itself. Output The first and the only line of the output should contain the answer to the problem. Examples Input 3 -1 -2 -3 Output 6 Input 5 -4 2 0 5 0 Output 11 Input 5 -1 10 -5 10 -2 Output 18 Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. In Berland it is the holiday of equality. In honor of the holiday the king decided to equalize the welfare of all citizens in Berland by the expense of the state treasury. Totally in Berland there are n citizens, the welfare of each of them is estimated as the integer in a_{i} burles (burle is the currency in Berland). You are the royal treasurer, which needs to count the minimum charges of the kingdom on the king's present. The king can only give money, he hasn't a power to take away them. -----Input----- The first line contains the integer n (1 ≤ n ≤ 100) — the number of citizens in the kingdom. The second line contains n integers a_1, a_2, ..., a_{n}, where a_{i} (0 ≤ a_{i} ≤ 10^6) — the welfare of the i-th citizen. -----Output----- In the only line print the integer S — the minimum number of burles which are had to spend. -----Examples----- Input 5 0 1 2 3 4 Output 10 Input 5 1 1 0 1 1 Output 1 Input 3 1 3 1 Output 4 Input 1 12 Output 0 -----Note----- In the first example if we add to the first citizen 4 burles, to the second 3, to the third 2 and to the fourth 1, then the welfare of all citizens will equal 4. In the second example it is enough to give one burle to the third citizen. In the third example it is necessary to give two burles to the first and the third citizens to make the welfare of citizens equal 3. In the fourth example it is possible to give nothing to everyone because all citizens have 12 burles. 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 prepared $n$ competitive programming problems. The topic of the $i$-th problem is $a_i$, and some problems' topics may coincide. Polycarp has to host several thematic contests. All problems in each contest should have the same topic, and all contests should have pairwise distinct topics. He may not use all the problems. It is possible that there are no contests for some topics. Polycarp wants to host competitions on consecutive days, one contest per day. Polycarp wants to host a set of contests in such a way that: number of problems in each contest is exactly twice as much as in the previous contest (one day ago), the first contest can contain arbitrary number of problems; the total number of problems in all the contests should be maximized. Your task is to calculate the maximum number of problems in the set of thematic contests. Note, that you should not maximize the number of contests. -----Input----- The first line of the input contains one integer $n$ ($1 \le n \le 2 \cdot 10^5$) — the number of problems Polycarp has prepared. The second line of the input contains $n$ integers $a_1, a_2, \dots, a_n$ ($1 \le a_i \le 10^9$) where $a_i$ is the topic of the $i$-th problem. -----Output----- Print one integer — the maximum number of problems in the set of thematic contests. -----Examples----- Input 18 2 1 2 10 2 10 10 2 2 1 10 10 10 10 1 1 10 10 Output 14 Input 10 6 6 6 3 6 1000000000 3 3 6 6 Output 9 Input 3 1337 1337 1337 Output 3 -----Note----- In the first example the optimal sequence of contests is: $2$ problems of the topic $1$, $4$ problems of the topic $2$, $8$ problems of the topic $10$. In the second example the optimal sequence of contests is: $3$ problems of the topic $3$, $6$ problems of the topic $6$. In the third example you can take all the problems with the topic $1337$ (the number of such problems is $3$ so the answer is $3$) and host a single contest. 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. Hongcow likes solving puzzles. One day, Hongcow finds two identical puzzle pieces, with the instructions "make a rectangle" next to them. The pieces can be described by an n by m grid of characters, where the character 'X' denotes a part of the puzzle and '.' denotes an empty part of the grid. It is guaranteed that the puzzle pieces are one 4-connected piece. See the input format and samples for the exact details on how a jigsaw piece will be specified. The puzzle pieces are very heavy, so Hongcow cannot rotate or flip the puzzle pieces. However, he is allowed to move them in any directions. The puzzle pieces also cannot overlap. You are given as input the description of one of the pieces. Determine if it is possible to make a rectangle from two identical copies of the given input. The rectangle should be solid, i.e. there should be no empty holes inside it or on its border. Keep in mind that Hongcow is not allowed to flip or rotate pieces and they cannot overlap, i.e. no two 'X' from different pieces can share the same position. -----Input----- The first line of input will contain two integers n and m (1 ≤ n, m ≤ 500), the dimensions of the puzzle piece. The next n lines will describe the jigsaw piece. Each line will have length m and will consist of characters '.' and 'X' only. 'X' corresponds to a part of the puzzle piece, '.' is an empty space. It is guaranteed there is at least one 'X' character in the input and that the 'X' characters form a 4-connected region. -----Output----- Output "YES" if it is possible for Hongcow to make a rectangle. Output "NO" otherwise. -----Examples----- Input 2 3 XXX XXX Output YES Input 2 2 .X XX Output NO Input 5 5 ..... ..X.. ..... ..... ..... Output YES -----Note----- For the first sample, one example of a rectangle we can form is as follows 111222 111222 For the second sample, it is impossible to put two of those pieces without rotating or flipping to form a rectangle. In the third sample, we can shift the first tile by one to the right, and then compose the following rectangle: ..... ..XX. ..... ..... ..... Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Monocarp wants to watch $n$ videos. Each video is only one minute long, but its size may be arbitrary. The $i$-th video has the size $a_i$ megabytes. All videos are published on the Internet. A video should be downloaded before it can be watched. Monocarp has poor Internet connection — it takes exactly $1$ minute to download $1$ megabyte of data, so it will require $a_i$ minutes to download the $i$-th video. Monocarp's computer has a hard disk of $m$ megabytes. The disk is used to store the downloaded videos. Once Monocarp starts the download of a video of size $s$, the $s$ megabytes are immediately reserved on a hard disk. If there are less than $s$ megabytes left, the download cannot be started until the required space is freed. Each single video can be stored on the hard disk, since $a_i \le m$ for all $i$. Once the download is started, it cannot be interrupted. It is not allowed to run two or more downloads in parallel. Once a video is fully downloaded to the hard disk, Monocarp can watch it. Watching each video takes exactly $1$ minute and does not occupy the Internet connection, so Monocarp can start downloading another video while watching the current one. When Monocarp finishes watching a video, he doesn't need it on the hard disk anymore, so he can delete the video, instantly freeing the space it occupied on a hard disk. Deleting a video takes negligible time. Monocarp wants to watch all $n$ videos as quickly as possible. The order of watching does not matter, since Monocarp needs to watch all of them anyway. Please calculate the minimum possible time required for that. -----Input----- The first line contains two integers $n$ and $m$ ($1 \le n \le 2 \cdot 10^5$; $1 \le m \le 10^9$) — the number of videos Monocarp wants to watch and the size of the hard disk, respectively. The second line contains $n$ integers $a_1, a_2, \dots, a_n$ ($1 \le a_i \le m$) — the sizes of the videos. -----Output----- Print one integer — the minimum time required to watch all $n$ videos. -----Examples----- Input 5 6 1 2 3 4 5 Output 16 Input 5 5 1 2 3 4 5 Output 17 Input 4 3 1 3 2 3 Output 12 -----Note----- None 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. Pablo Squarson is a well-known cubism artist. This year's theme for Pablo Squarson is "Squares". Today we are visiting his studio to see how his masterpieces are given birth. At the center of his studio, there is a huuuuuge table and beside it are many, many squares of the same size. Pablo Squarson puts one of the squares on the table. Then he places some other squares on the table in sequence. It seems his methodical nature forces him to place each square side by side to the one that he already placed on, with machine-like precision. Oh! The first piece of artwork is done. Pablo Squarson seems satisfied with it. Look at his happy face. Oh, what's wrong with Pablo? He is tearing his hair! Oh, I see. He wants to find a box that fits the new piece of work but he has trouble figuring out its size. Let's help him! Your mission is to write a program that takes instructions that record how Pablo made a piece of his artwork and computes its width and height. It is known that the size of each square is 1. You may assume that Pablo does not put a square on another. I hear someone murmured "A smaller box will do". No, poor Pablo, shaking his head, is grumbling "My square style does not seem to be understood by illiterates". <image> Input The input consists of a number of datasets. Each dataset represents the way Pablo made a piece of his artwork. The format of a dataset is as follows. > N n1 d1 n2 d2 ... nN-1 dN-1 The first line contains the number of squares (= N) used to make the piece of artwork. The number is a positive integer and is smaller than 200. The remaining (N-1) lines in the dataset are square placement instructions. The line "ni di" indicates placement of the square numbered i (≤ N-1). The rules of numbering squares are as follows. The first square is numbered "zero". Subsequently placed squares are numbered 1, 2, ..., (N-1). Note that the input does not give any placement instruction to the first square, which is numbered zero. A square placement instruction for the square numbered i, namely "ni di", directs it to be placed next to the one that is numbered ni, towards the direction given by di, which denotes leftward (= 0), downward (= 1), rightward (= 2), and upward (= 3). For example, pieces of artwork corresponding to the four datasets shown in Sample Input are depicted below. Squares are labeled by their numbers. <image> The end of the input is indicated by a line that contains a single zero. Output For each dataset, output a line that contains the width and the height of the piece of artwork as decimal numbers, separated by a space. Each line should not contain any other characters. Sample Input 1 5 0 0 0 1 0 2 0 3 12 0 0 1 0 2 0 3 1 4 1 5 1 6 2 7 2 8 2 9 3 10 3 10 0 2 1 2 2 2 3 2 2 1 5 1 6 1 7 1 8 1 0 Output for the Sample Input 1 1 3 3 4 4 5 6 Example Input 1 5 0 0 0 1 0 2 0 3 12 0 0 1 0 2 0 3 1 4 1 5 1 6 2 7 2 8 2 9 3 10 3 10 0 2 1 2 2 2 3 2 2 1 5 1 6 1 7 1 8 1 0 Output 1 1 3 3 4 4 5 6 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 appointed director of a famous concert hall, to save it from bankruptcy. The hall is very popular, and receives many requests to use its two fine rooms, but unfortunately the previous director was not very efficient, and it has been losing money for many years. The two rooms are of the same size and arrangement. Therefore, each applicant wishing to hold a concert asks for a room without specifying which. Each room can be used for only one concert per day. In order to make more money, you have decided to abandon the previous fixed price policy, and rather let applicants specify the price they are ready to pay. Each application shall specify a period [i, j] and an asking price w, where i and j are respectively the first and last days of the period (1 ≤ i ≤ j ≤ 365), and w is a positive integer in yen, indicating the amount the applicant is willing to pay to use a room for the whole period. You have received applications for the next year, and you should now choose the applications you will accept. Each application should be either accepted for its whole period or completely rejected. Each concert should use the same room during the whole applied period. Considering the dire economic situation of the concert hall, artistic quality is to be ignored, and you should just try to maximize the total income for the whole year by accepting the most profitable applications. Input The input has multiple data sets, each starting with a line consisting of a single integer n, the number of applications in the data set. Then, it is followed by n lines, each of which represents one application with a period [i, j] and an asking price w yen in the following format. i j w A line containing a single zero indicates the end of the input. The maximum number of applications in a data set is one thousand, and the maximum asking price is one million yen. Output For each data set, print a single line containing an integer, the maximum total income in yen for the data set. Example Input 4 1 2 10 2 3 10 3 3 10 1 3 10 6 1 20 1000 3 25 10000 5 15 5000 22 300 5500 10 295 9000 7 7 6000 8 32 251 2261 123 281 1339 211 235 5641 162 217 7273 22 139 7851 194 198 9190 119 274 878 122 173 8640 0 Output 30 25500 38595 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 bulbs arranged on a number line, numbered 1 to N from left to right. Bulb i is at coordinate i. Each bulb has a non-negative integer parameter called intensity. When there is a bulb of intensity d at coordinate x, the bulb illuminates the segment from coordinate x-d-0.5 to x+d+0.5. Initially, the intensity of Bulb i is A_i. We will now do the following operation K times in a row: * For each integer i between 1 and N (inclusive), let B_i be the number of bulbs illuminating coordinate i. Then, change the intensity of each bulb i to B_i. Find the intensity of each bulb after the K operations. Constraints * 1 \leq N \leq 2 \times 10^5 * 1 \leq K \leq 2 \times 10^5 * 0 \leq A_i \leq N Input Input is given from Standard Input in the following format: N K A_1 A_2 \ldots A_N Output Print the intensity A{'}_i of each bulb i after the K operations to Standard Output in the following format: A{'}_1 A{'}_2 \ldots A{'}_N Output Print the intensity A{'}_i of each bulb i after the K operations to Standard Output in the following format: A{'}_1 A{'}_2 \ldots A{'}_N Examples Input 5 1 1 0 0 1 0 Output 1 2 2 1 2 Input 5 2 1 0 0 1 0 Output 3 3 4 4 3 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 train going from Station A to Station B that costs X yen (the currency of Japan). Also, there is a bus going from Station B to Station C that costs Y yen. Joisino got a special ticket. With this ticket, she can take the bus for half the fare if she travels from Station A to Station B by train and then travels from Station B to Station C by bus. How much does it cost to travel from Station A to Station C if she uses this ticket? -----Constraints----- - 1 \leq X,Y \leq 100 - Y is an even number. - All values in input are integers. -----Input----- Input is given from Standard Input in the following format: X Y -----Output----- If it costs x yen to travel from Station A to Station C, print x. -----Sample Input----- 81 58 -----Sample Output----- 110 - The train fare is 81 yen. - The train fare is 58 ⁄ 2=29 yen with the 50% discount. Thus, it costs 110 yen to travel from Station A to Station C. Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. You are given a string S of length N consisting of ( and ). Your task is to insert some number of ( and ) into S to obtain a correct bracket sequence. Here, a correct bracket sequence is defined as follows: - () is a correct bracket sequence. - If X is a correct bracket sequence, the concatenation of (, X and ) in this order is also a correct bracket sequence. - If X and Y are correct bracket sequences, the concatenation of X and Y in this order is also a correct bracket sequence. - Every correct bracket sequence can be derived from the rules above. Find the shortest correct bracket sequence that can be obtained. If there is more than one such sequence, find the lexicographically smallest one. -----Constraints----- - The length of S is N. - 1 ≤ N ≤ 100 - S consists of ( and ). -----Input----- Input is given from Standard Input in the following format: N S -----Output----- Print the lexicographically smallest string among the shortest correct bracket sequences that can be obtained by inserting some number of ( and ) into S. -----Sample Input----- 3 ()) -----Sample Output----- (()) 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. Sig has built his own keyboard. Designed for ultimate simplicity, this keyboard only has 3 keys on it: the 0 key, the 1 key and the backspace key. To begin with, he is using a plain text editor with this keyboard. This editor always displays one string (possibly empty). Just after the editor is launched, this string is empty. When each key on the keyboard is pressed, the following changes occur to the string: - The 0 key: a letter 0 will be inserted to the right of the string. - The 1 key: a letter 1 will be inserted to the right of the string. - The backspace key: if the string is empty, nothing happens. Otherwise, the rightmost letter of the string is deleted. Sig has launched the editor, and pressed these keys N times in total. As a result, the editor displays a string s. Find the number of such ways to press the keys, modulo 10^9 + 7. -----Constraints----- - 1 ≦ N ≦ 5000 - 1 ≦ \|s\| ≦ N - s consists of the letters 0 and 1. -----Partial Score----- - 400 points will be awarded for passing the test set satisfying 1 ≦ N ≦ 300. -----Input----- The input is given from Standard Input in the following format: N s -----Output----- Print the number of the ways to press the keys N times in total such that the editor displays the string s in the end, modulo 10^9+7. -----Sample Input----- 3 0 -----Sample Output----- 5 We will denote the backspace key by B. The following 5 ways to press the keys will cause the editor to display the string 0 in the end: 00B, 01B, 0B0, 1B0, BB0. In the last way, nothing will happen when the backspace key is pressed. 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 C loves number «3» very much. He loves all things about it. Now he is playing a game on a chessboard of size $n \times m$. The cell in the $x$-th row and in the $y$-th column is called $(x,y)$. Initially, The chessboard is empty. Each time, he places two chessmen on two different empty cells, the Manhattan distance between which is exactly $3$. The Manhattan distance between two cells $(x_i,y_i)$ and $(x_j,y_j)$ is defined as $\|x_i-x_j\|+\|y_i-y_j\|$. He want to place as many chessmen as possible on the chessboard. Please help him find the maximum number of chessmen he can place. -----Input----- A single line contains two integers $n$ and $m$ ($1 \leq n,m \leq 10^9$) — the number of rows and the number of columns of the chessboard. -----Output----- Print one integer — the maximum number of chessmen Little C can place. -----Examples----- Input 2 2 Output 0 Input 3 3 Output 8 -----Note----- In the first example, the Manhattan distance between any two cells is smaller than $3$, so the answer is $0$. In the second example, a possible solution is $(1,1)(3,2)$, $(1,2)(3,3)$, $(2,1)(1,3)$, $(3,1)(2,3)$. Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Iahub accidentally discovered a secret lab. He found there n devices ordered in a line, numbered from 1 to n from left to right. Each device i (1 ≤ i ≤ n) can create either ai units of matter or ai units of antimatter. Iahub wants to choose some contiguous subarray of devices in the lab, specify the production mode for each of them (produce matter or antimatter) and finally take a photo of it. However he will be successful only if the amounts of matter and antimatter produced in the selected subarray will be the same (otherwise there would be overflowing matter or antimatter in the photo). You are requested to compute the number of different ways Iahub can successful take a photo. A photo is different than another if it represents another subarray, or if at least one device of the subarray is set to produce matter in one of the photos and antimatter in the other one. Input The first line contains an integer n (1 ≤ n ≤ 1000). The second line contains n integers a1, a2, ..., an (1 ≤ ai ≤ 1000). The sum a1 + a2 + ... + an will be less than or equal to 10000. Output Output a single integer, the number of ways Iahub can take a photo, modulo 1000000007 (109 + 7). Examples Input 4 1 1 1 1 Output 12 Note The possible photos are [1+, 2-], [1-, 2+], [2+, 3-], [2-, 3+], [3+, 4-], [3-, 4+], [1+, 2+, 3-, 4-], [1+, 2-, 3+, 4-], [1+, 2-, 3-, 4+], [1-, 2+, 3+, 4-], [1-, 2+, 3-, 4+] and [1-, 2-, 3+, 4+], where "i+" means that the i-th element produces matter, and "i-" means that the i-th element produces antimatter. 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. Drazil is playing a math game with Varda. Let's define <image> for positive integer x as a product of factorials of its digits. For example, <image>. First, they choose a decimal number a consisting of n digits that contains at least one digit larger than 1. This number may possibly start with leading zeroes. Then they should find maximum positive number x satisfying following two conditions: 1. x doesn't contain neither digit 0 nor digit 1. 2. <image> = <image>. Help friends find such number. Input The first line contains an integer n (1 ≤ n ≤ 15) — the number of digits in a. The second line contains n digits of a. There is at least one digit in a that is larger than 1. Number a may possibly contain leading zeroes. Output Output a maximum possible integer satisfying the conditions above. There should be no zeroes and ones in this number decimal representation. Examples Input 4 1234 Output 33222 Input 3 555 Output 555 Note In the first case, <image> Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. One day three best friends Petya, Vasya and Tonya decided to form a team and take part in programming contests. Participants are usually offered several problems during programming contests. Long before the start the friends decided that they will implement a problem if at least two of them are sure about the solution. Otherwise, the friends won't write the problem's solution. This contest offers n problems to the participants. For each problem we know, which friend is sure about the solution. Help the friends find the number of problems for which they will write a solution. Input The first input line contains a single integer n (1 ≤ n ≤ 1000) — the number of problems in the contest. Then n lines contain three integers each, each integer is either 0 or 1. If the first number in the line equals 1, then Petya is sure about the problem's solution, otherwise he isn't sure. The second number shows Vasya's view on the solution, the third number shows Tonya's view. The numbers on the lines are separated by spaces. Output Print a single integer — the number of problems the friends will implement on the contest. Examples Input 3 1 1 0 1 1 1 1 0 0 Output 2 Input 2 1 0 0 0 1 1 Output 1 Note In the first sample Petya and Vasya are sure that they know how to solve the first problem and all three of them know how to solve the second problem. That means that they will write solutions for these problems. Only Petya is sure about the solution for the third problem, but that isn't enough, so the friends won't take it. In the second sample the friends will only implement the second problem, as Vasya and Tonya are sure about the solution. 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 locked in a room with a door that has a keypad with 10 keys corresponding to digits from 0 to 9. To escape from the room, you need to enter a correct code. You also have a sequence of digits. Some keys on the keypad have fingerprints. You believe the correct code is the longest not necessarily contiguous subsequence of the sequence you have that only contains digits with fingerprints on the corresponding keys. Find such code. -----Input----- The first line contains two integers $n$ and $m$ ($1 \le n, m \le 10$) representing the number of digits in the sequence you have and the number of keys on the keypad that have fingerprints. The next line contains $n$ distinct space-separated integers $x_1, x_2, \ldots, x_n$ ($0 \le x_i \le 9$) representing the sequence. The next line contains $m$ distinct space-separated integers $y_1, y_2, \ldots, y_m$ ($0 \le y_i \le 9$) — the keys with fingerprints. -----Output----- In a single line print a space-separated sequence of integers representing the code. If the resulting sequence is empty, both printing nothing and printing a single line break is acceptable. -----Examples----- Input 7 3 3 5 7 1 6 2 8 1 2 7 Output 7 1 2 Input 4 4 3 4 1 0 0 1 7 9 Output 1 0 -----Note----- In the first example, the only digits with fingerprints are $1$, $2$ and $7$. All three of them appear in the sequence you know, $7$ first, then $1$ and then $2$. Therefore the output is 7 1 2. Note that the order is important, and shall be the same as the order in the original sequence. In the second example digits $0$, $1$, $7$ and $9$ have fingerprints, however only $0$ and $1$ appear in the original sequence. $1$ appears earlier, so the output is 1 0. Again, the order is important. 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 "BerCorp" company has got n employees. These employees can use m approved official languages for the formal correspondence. The languages are numbered with integers from 1 to m. For each employee we have the list of languages, which he knows. This list could be empty, i. e. an employee may know no official languages. But the employees are willing to learn any number of official languages, as long as the company pays their lessons. A study course in one language for one employee costs 1 berdollar. Find the minimum sum of money the company needs to spend so as any employee could correspond to any other one (their correspondence can be indirect, i. e. other employees can help out translating). -----Input----- The first line contains two integers n and m (2 ≤ n, m ≤ 100) — the number of employees and the number of languages. Then n lines follow — each employee's language list. At the beginning of the i-th line is integer k_{i} (0 ≤ k_{i} ≤ m) — the number of languages the i-th employee knows. Next, the i-th line contains k_{i} integers — a_{ij} (1 ≤ a_{ij} ≤ m) — the identifiers of languages the i-th employee knows. It is guaranteed that all the identifiers in one list are distinct. Note that an employee may know zero languages. The numbers in the lines are separated by single spaces. -----Output----- Print a single integer — the minimum amount of money to pay so that in the end every employee could write a letter to every other one (other employees can help out translating). -----Examples----- Input 5 5 1 2 2 2 3 2 3 4 2 4 5 1 5 Output 0 Input 8 7 0 3 1 2 3 1 1 2 5 4 2 6 7 1 3 2 7 4 1 1 Output 2 Input 2 2 1 2 0 Output 1 -----Note----- In the second sample the employee 1 can learn language 2, and employee 8 can learn language 4. In the third sample employee 2 must learn language 2. 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. Theofanis decided to visit his uncle's farm. There are $s$ animals and $n$ animal pens on the farm. For utility purpose, animal pens are constructed in one row. Uncle told Theofanis that a farm is lucky if you can distribute all animals in all pens in such a way that there are no empty pens and there is at least one continuous segment of pens that has exactly $k$ animals in total. Moreover, a farm is ideal if it's lucky for any distribution without empty pens. Neither Theofanis nor his uncle knows if their farm is ideal or not. Can you help them to figure it out? -----Input----- The first line contains a single integer $t$ ($1 \le t \le 10^5$) — the number of test cases. The first and only line of each test case contains three integers $s$, $n$, and $k$ ($1 \le s, n, k \le 10^{18}$; $n \le s$). -----Output----- For each test case, print YES (case-insensitive), if the farm is ideal, or NO (case-insensitive) otherwise. -----Examples----- Input 4 1 1 1 1 1 2 100 50 200 56220 47258 14497 Output YES NO NO YES -----Note----- For the first and the second test case, the only possible combination is $[1]$ so there always will be a subsegment with $1$ animal but not with $2$ animals. 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 two infinite sources of water: hot water of temperature $h$; cold water of temperature $c$ ($c < h$). You perform the following procedure of alternating moves: take one cup of the hot water and pour it into an infinitely deep barrel; take one cup of the cold water and pour it into an infinitely deep barrel; take one cup of the hot water $\dots$ and so on $\dots$ Note that you always start with the cup of hot water. The barrel is initially empty. You have to pour at least one cup into the barrel. The water temperature in the barrel is an average of the temperatures of the poured cups. You want to achieve a temperature as close as possible to $t$. So if the temperature in the barrel is $t_b$, then the absolute difference of $t_b$ and $t$ ($\|t_b - t\|$) should be as small as possible. How many cups should you pour into the barrel, so that the temperature in it is as close as possible to $t$? If there are multiple answers with the minimum absolute difference, then print the smallest of them. -----Input----- The first line contains a single integer $T$ ($1 \le T \le 3 \cdot 10^4$) — the number of testcases. Each of the next $T$ lines contains three integers $h$, $c$ and $t$ ($1 \le c < h \le 10^6$; $c \le t \le h$) — the temperature of the hot water, the temperature of the cold water and the desired temperature in the barrel. -----Output----- For each testcase print a single positive integer — the minimum number of cups required to be poured into the barrel to achieve the closest temperature to $t$. -----Example----- Input 3 30 10 20 41 15 30 18 13 18 Output 2 7 1 -----Note----- In the first testcase the temperature after $2$ poured cups: $1$ hot and $1$ cold is exactly $20$. So that is the closest we can achieve. In the second testcase the temperature after $7$ poured cups: $4$ hot and $3$ cold is about $29.857$. Pouring more water won't get us closer to $t$ than that. In the third testcase the temperature after $1$ poured cup: $1$ hot is $18$. That's exactly equal to $t$. Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. This is an interactive problem. Refer to the Interaction section below for better understanding. Ithea and Chtholly want to play a game in order to determine who can use the kitchen tonight. <image> Initially, Ithea puts n clear sheets of paper in a line. They are numbered from 1 to n from left to right. This game will go on for m rounds. In each round, Ithea will give Chtholly an integer between 1 and c, and Chtholly needs to choose one of the sheets to write down this number (if there is already a number before, she will erase the original one and replace it with the new one). Chtholly wins if, at any time, all the sheets are filled with a number and the n numbers are in non-decreasing order looking from left to right from sheet 1 to sheet n, and if after m rounds she still doesn't win, she loses the game. Chtholly really wants to win the game as she wants to cook something for Willem. But she doesn't know how to win the game. So Chtholly finds you, and your task is to write a program to receive numbers that Ithea gives Chtholly and help her make the decision on which sheet of paper write this number. Input The first line contains 3 integers n, m and c (<image>, <image> means <image> rounded up) — the number of sheets, the number of rounds and the largest possible number Ithea can give to Chtholly respectively. The remaining parts of input are given throughout the interaction process. Interaction In each round, your program needs to read one line containing a single integer pi (1 ≤ pi ≤ c), indicating the number given to Chtholly. Your program should then output a line containing an integer between 1 and n, indicating the number of sheet to write down this number in. After outputting each line, don't forget to flush the output. For example: * fflush(stdout) in C/C++; * System.out.flush() in Java; * sys.stdout.flush() in Python; * flush(output) in Pascal; * See the documentation for other languages. If Chtholly wins at the end of a round, no more input will become available and your program should terminate normally. It can be shown that under the constraints, it's always possible for Chtholly to win the game. Example Input 2 4 4 2 1 3 Output 1 2 2 Note In the example, Chtholly initially knew there were 2 sheets, 4 rounds and each number was between 1 and 4. She then received a 2 and decided to write it in the 1st sheet. Then she received a 1 and wrote it in the 2nd sheet. At last, she received a 3 and replaced 1 with 3 in the 2nd sheet. At this time all the sheets were filled with a number and they were non-decreasing, so she won the game. Note that it is required that your program terminate immediately after Chtholly wins and do not read numbers from the input for the remaining rounds. If not, undefined behaviour may arise and it won't be sure whether your program will be accepted or rejected. Also because of this, please be careful when hacking others' codes. In the sample, Chtholly won the game after the 3rd round, so it is required that your program doesn't read the number of the remaining 4th round. The input format for hacking: * The first line contains 3 integers n, m and c; * The following m lines each contains an integer between 1 and c, indicating the number given to Chtholly in each round. 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. ### Background I was reading a [book](http://www.amazon.co.uk/Things-Make-Do-Fourth-Dimension/dp/1846147646/) recently, "Things to Make and Do in the Fourth Dimension" by comedian and mathematician Matt Parker, and in the first chapter of the book Matt talks about problems he likes to solve in his head to take his mind off the fact that he is in his dentist's chair, we've all been there! The problem he talks about relates to polydivisible numbers, and I thought a kata should be written on the subject as it's quite interesting. (Well it's interesting to me, so there!) ### Polydivisib... huh what? So what are they? A polydivisible number is divisible in an unusual way. The first digit is cleanly divisible by `1`, the first two digits are cleanly divisible by `2`, the first three by `3` and so on. The interesting thing about polydivisiblity is that it relates to the underlying number, but not the base it is written in, so if aliens came to Earth and used base `23` (`11` fingers on one hand and `12` on the other), no matter what squiggles they use to write numbers, they would find the same numbers polydivisible! ### Polydivisibilty Example: Let's do a worked example to clear up any questions ... Starting wih the number `1,232` in base `10` then: ``` 1232 1 /1 = 1 Yay! 12 /2 = 6 Yay! 123 /3 = 41 Yay! 1232 /4 = 308 Yay! ``` Thus `1,232` is a polydivisible number in base `4` and above. However starting wih the number `123,220` and using base `10` then: ``` 123220 1 /1 = 1 Yay! 12 /2 = 6 Yay! 123 /3 = 41 Yay! 1232 /4 = 308 Yay! 12322 /5 = 2464.4 Oh no, that's not a round number! 123220 /6 = 220536.333r Oh no, that's not a round number! ``` Thus `123,220` is not a polydivisible base 10 number, but what about in another base? Again starting wih the number `123,220` and using base `6` then: ``` base 6 base 10 1 = 1 -> 1 /1 = 1 Yay! 12 = 8 -> 8 /2 = 4 Yay! 123 = 51 -> 51 /3 = 17 Yay! 1232 = 308 -> 308 /4 = 77 Yay! 12322 = 1850 -> 1850 /5 = 370 Yay! 123220 = 11100 -> 11100 /6 = 1850 Yay! ``` Thus `123,220` is a polydivisible base `6` number (and a polydivisible base `10` number when converted to `11100` in base `10`). ### Kata In this kata you must implement two methods: `is_polydivisible(n, b)` and `get_polydivisible(n, b)`. The first `is_polydivisible(n, b)` will return `True` if `n` is polydivisible in base `b` or `False` if not. The second `get_polydivisible(n, b)` will return the `n`th polydivisible number using base `b`, the first polydivisible number is of course always `0`. You can assume that all inputs are valid. ```if:haskell All necessary arithmetic can be done in `Int` range. ``` ### Kata Examples: ```python is_polydivisible("1232", 10) # => True is_polydivisible("123220", 10) # => False is_polydivisible("123220", 6) # => True get_polydivisible(22, 10) # => "32" get_polydivisible(22, 16) # => "1A" get_polydivisible(42, 16) # => "42" ``` #### A Note on Bases The maximum base used is base `62`, and uses characters in the following order `[0-9][A-Z][a-z]` to denote its digits, base `n` will use the first `n` characters of this sequence. ```if-not:haskell A constant CHARS has been declared with this sequence for you. ``` 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 Ancient Kingdom of Snuke, there was a pyramid to strengthen the authority of Takahashi, the president of AtCoder Inc. The pyramid had center coordinates (C_X, C_Y) and height H. The altitude of coordinates (X, Y) is max(H - \|X - C_X\| - \|Y - C_Y\|, 0). Aoki, an explorer, conducted a survey to identify the center coordinates and height of this pyramid. As a result, he obtained the following information: - C_X, C_Y was integers between 0 and 100 (inclusive), and H was an integer not less than 1. - Additionally, he obtained N pieces of information. The i-th of them is: "the altitude of point (x_i, y_i) is h_i." This was enough to identify the center coordinates and the height of the pyramid. Find these values with the clues above. -----Constraints----- - N is an integer between 1 and 100 (inclusive). - x_i and y_i are integers between 0 and 100 (inclusive). - h_i is an integer between 0 and 10^9 (inclusive). - The N coordinates (x_1, y_1), (x_2, y_2), (x_3, y_3), ..., (x_N, y_N) are all different. - The center coordinates and the height of the pyramid can be uniquely identified. -----Input----- Input is given from Standard Input in the following format: N x_1 y_1 h_1 x_2 y_2 h_2 x_3 y_3 h_3 : x_N y_N h_N -----Output----- Print values C_X, C_Y and H representing the center coordinates and the height of the pyramid in one line, with spaces in between. -----Sample Input----- 4 2 3 5 2 1 5 1 2 5 3 2 5 -----Sample Output----- 2 2 6 In this case, the center coordinates and the height can be identified as (2, 2) and 6. Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. One of the most important products of the R1 company is a popular @r1.com mail service. The R1 mailboxes receive and send millions of emails every day. Today, the online news thundered with terrible information. The R1 database crashed and almost no data could be saved except for one big string. The developers assume that the string contains the letters of some users of the R1 mail. Recovering letters is a tedious mostly manual work. So before you start this process, it was decided to estimate the difficulty of recovering. Namely, we need to calculate the number of different substrings of the saved string that form correct e-mail addresses. We assume that valid addresses are only the e-mail addresses which meet the following criteria: * the address should begin with a non-empty sequence of letters, numbers, characters '_', starting with a letter; * then must go character '@'; * then must go a non-empty sequence of letters or numbers; * then must go character '.'; * the address must end with a non-empty sequence of letters. You got lucky again and the job was entrusted to you! Please note that the substring is several consecutive characters in a string. Two substrings, one consisting of the characters of the string with numbers l1, l1 + 1, l1 + 2, ..., r1 and the other one consisting of the characters of the string with numbers l2, l2 + 1, l2 + 2, ..., r2, are considered distinct if l1 ≠ l2 or r1 ≠ r2. Input The first and the only line contains the sequence of characters s1s2... sn (1 ≤ n ≤ 106) — the saved string. It is guaranteed that the given string contains only small English letters, digits and characters '.', '_', '@'. Output Print in a single line the number of substrings that are valid e-mail addresses. Examples Input gerald.agapov1991@gmail.com Output 18 Input x@x.x@x.x_e_@r1.com Output 8 Input a___@1.r Output 1 Input .asd123__..@ Output 0 Note In the first test case all the substrings that are correct e-mail addresses begin from one of the letters of the word agapov and end in one of the letters of the word com. In the second test case note that the e-mail x@x.x is considered twice in the answer. Note that in this example the e-mail entries overlap inside the string. 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. Vanya has a scales for weighing loads and weights of masses w^0, w^1, w^2, ..., w^100 grams where w is some integer not less than 2 (exactly one weight of each nominal value). Vanya wonders whether he can weight an item with mass m using the given weights, if the weights can be put on both pans of the scales. Formally speaking, your task is to determine whether it is possible to place an item of mass m and some weights on the left pan of the scales, and some weights on the right pan of the scales so that the pans of the scales were in balance. -----Input----- The first line contains two integers w, m (2 ≤ w ≤ 10^9, 1 ≤ m ≤ 10^9) — the number defining the masses of the weights and the mass of the item. -----Output----- Print word 'YES' if the item can be weighted and 'NO' if it cannot. -----Examples----- Input 3 7 Output YES Input 100 99 Output YES Input 100 50 Output NO -----Note----- Note to the first sample test. One pan can have an item of mass 7 and a weight of mass 3, and the second pan can have two weights of masses 9 and 1, correspondingly. Then 7 + 3 = 9 + 1. Note to the second sample test. One pan of the scales can have an item of mass 99 and the weight of mass 1, and the second pan can have the weight of mass 100. Note to the third sample test. It is impossible to measure the weight of the item in the manner described in the input. Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Vasya has a sequence of cubes and exactly one integer is written on each cube. Vasya exhibited all his cubes in a row. So the sequence of numbers written on the cubes in the order from the left to the right equals to a_1, a_2, ..., a_{n}. While Vasya was walking, his little brother Stepan played with Vasya's cubes and changed their order, so now the sequence of numbers written on the cubes became equal to b_1, b_2, ..., b_{n}. Stepan said that he swapped only cubes which where on the positions between l and r, inclusive, and did not remove or add any other cubes (i. e. he said that he reordered cubes between positions l and r, inclusive, in some way). Your task is to determine if it is possible that Stepan said the truth, or it is guaranteed that Stepan deceived his brother. -----Input----- The first line contains three integers n, l, r (1 ≤ n ≤ 10^5, 1 ≤ l ≤ r ≤ n) — the number of Vasya's cubes and the positions told by Stepan. The second line contains the sequence a_1, a_2, ..., a_{n} (1 ≤ a_{i} ≤ n) — the sequence of integers written on cubes in the Vasya's order. The third line contains the sequence b_1, b_2, ..., b_{n} (1 ≤ b_{i} ≤ n) — the sequence of integers written on cubes after Stepan rearranged their order. It is guaranteed that Stepan did not remove or add other cubes, he only rearranged Vasya's cubes. -----Output----- Print "LIE" (without quotes) if it is guaranteed that Stepan deceived his brother. In the other case, print "TRUTH" (without quotes). -----Examples----- Input 5 2 4 3 4 2 3 1 3 2 3 4 1 Output TRUTH Input 3 1 2 1 2 3 3 1 2 Output LIE Input 4 2 4 1 1 1 1 1 1 1 1 Output TRUTH -----Note----- In the first example there is a situation when Stepan said the truth. Initially the sequence of integers on the cubes was equal to [3, 4, 2, 3, 1]. Stepan could at first swap cubes on positions 2 and 3 (after that the sequence of integers on cubes became equal to [3, 2, 4, 3, 1]), and then swap cubes in positions 3 and 4 (after that the sequence of integers on cubes became equal to [3, 2, 3, 4, 1]). In the second example it is not possible that Stepan said truth because he said that he swapped cubes only between positions 1 and 2, but we can see that it is guaranteed that he changed the position of the cube which was on the position 3 at first. So it is guaranteed that Stepan deceived his brother. In the third example for any values l and r there is a situation when Stepan said the truth. 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. Bob is an avid fan of the video game "League of Leesins", and today he celebrates as the League of Leesins World Championship comes to an end! The tournament consisted of $n$ ($n \ge 5$) teams around the world. Before the tournament starts, Bob has made a prediction of the rankings of each team, from $1$-st to $n$-th. After the final, he compared the prediction with the actual result and found out that the $i$-th team according to his prediction ended up at the $p_i$-th position ($1 \le p_i \le n$, all $p_i$ are unique). In other words, $p$ is a permutation of $1, 2, \dots, n$. As Bob's favorite League player is the famous "3ga", he decided to write down every $3$ consecutive elements of the permutation $p$. Formally, Bob created an array $q$ of $n-2$ triples, where $q_i = (p_i, p_{i+1}, p_{i+2})$ for each $1 \le i \le n-2$. Bob was very proud of his array, so he showed it to his friend Alice. After learning of Bob's array, Alice declared that she could retrieve the permutation $p$ even if Bob rearranges the elements of $q$ and the elements within each triple. Of course, Bob did not believe in such magic, so he did just the same as above to see Alice's respond. For example, if $n = 5$ and $p = [1, 4, 2, 3, 5]$, then the original array $q$ will be $[(1, 4, 2), (4, 2, 3), (2, 3, 5)]$. Bob can then rearrange the numbers within each triple and the positions of the triples to get $[(4, 3, 2), (2, 3, 5), (4, 1, 2)]$. Note that $[(1, 4, 2), (4, 2, 2), (3, 3, 5)]$ is not a valid rearrangement of $q$, as Bob is not allowed to swap numbers belong to different triples. As Alice's friend, you know for sure that Alice was just trying to show off, so you decided to save her some face by giving her any permutation $p$ that is consistent with the array $q$ she was given. -----Input----- The first line contains a single integer $n$ ($5 \le n \le 10^5$) — the size of permutation $p$. The $i$-th of the next $n-2$ lines contains $3$ integers $q_{i, 1}$, $q_{i, 2}$, $q_{i, 3}$ ($1 \le q_{i, j} \le n$) — the elements of the $i$-th triple of the rearranged (shuffled) array $q_i$, in random order. Remember, that the numbers within each triple can be rearranged and also the positions of the triples can be rearranged. It is guaranteed that there is at least one permutation $p$ that is consistent with the input. -----Output----- Print $n$ distinct integers $p_1, p_2, \ldots, p_n$ ($1 \le p_i \le n$) such that $p$ is consistent with array $q$. If there are multiple answers, print any. -----Example----- Input 5 4 3 2 2 3 5 4 1 2 Output 1 4 2 3 5 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 wants to cook a soup. To do it, he needs to buy exactly $n$ liters of water. There are only two types of water bottles in the nearby shop — $1$-liter bottles and $2$-liter bottles. There are infinitely many bottles of these two types in the shop. The bottle of the first type costs $a$ burles and the bottle of the second type costs $b$ burles correspondingly. Polycarp wants to spend as few money as possible. Your task is to find the minimum amount of money (in burles) Polycarp needs to buy exactly $n$ liters of water in the nearby shop if the bottle of the first type costs $a$ burles and the bottle of the second type costs $b$ burles. You also have to answer $q$ independent queries. -----Input----- The first line of the input contains one integer $q$ ($1 \le q \le 500$) — the number of queries. The next $n$ lines contain queries. The $i$-th query is given as three space-separated integers $n_i$, $a_i$ and $b_i$ ($1 \le n_i \le 10^{12}, 1 \le a_i, b_i \le 1000$) — how many liters Polycarp needs in the $i$-th query, the cost (in burles) of the bottle of the first type in the $i$-th query and the cost (in burles) of the bottle of the second type in the $i$-th query, respectively. -----Output----- Print $q$ integers. The $i$-th integer should be equal to the minimum amount of money (in burles) Polycarp needs to buy exactly $n_i$ liters of water in the nearby shop if the bottle of the first type costs $a_i$ burles and the bottle of the second type costs $b_i$ burles. -----Example----- Input 4 10 1 3 7 3 2 1 1000 1 1000000000000 42 88 Output 10 9 1000 42000000000000 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 is a user of a site that hosts programming contests. When a user competes in a contest, the rating of the user (not necessarily an integer) changes according to the performance of the user, as follows: - Let the current rating of the user be a. - Suppose that the performance of the user in the contest is b. - Then, the new rating of the user will be the avarage of a and b. For example, if a user with rating 1 competes in a contest and gives performance 1000, his/her new rating will be 500.5, the average of 1 and 1000. Takahashi's current rating is R, and he wants his rating to be exactly G after the next contest. Find the performance required to achieve it. -----Constraints----- - 0 \leq R, G \leq 4500 - All input values are integers. -----Input----- Input is given from Standard Input in the following format: R G -----Output----- Print the performance required to achieve the objective. -----Sample Input----- 2002 2017 -----Sample Output----- 2032 Takahashi's current rating is 2002. If his performance in the contest is 2032, his rating will be the average of 2002 and 2032, which is equal to the desired rating, 2017. 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 frog is initially at position $0$ on the number line. The frog has two positive integers $a$ and $b$. From a position $k$, it can either jump to position $k+a$ or $k-b$. Let $f(x)$ be the number of distinct integers the frog can reach if it never jumps on an integer outside the interval $[0, x]$. The frog doesn't need to visit all these integers in one trip, that is, an integer is counted if the frog can somehow reach it if it starts from $0$. Given an integer $m$, find $\sum_{i=0}^{m} f(i)$. That is, find the sum of all $f(i)$ for $i$ from $0$ to $m$. -----Input----- The first line contains three integers $m, a, b$ ($1 \leq m \leq 10^9, 1 \leq a,b \leq 10^5$). -----Output----- Print a single integer, the desired sum. -----Examples----- Input 7 5 3 Output 19 Input 1000000000 1 2019 Output 500000001500000001 Input 100 100000 1 Output 101 Input 6 4 5 Output 10 -----Note----- In the first example, we must find $f(0)+f(1)+\ldots+f(7)$. We have $f(0) = 1, f(1) = 1, f(2) = 1, f(3) = 1, f(4) = 1, f(5) = 3, f(6) = 3, f(7) = 8$. The sum of these values is $19$. In the second example, we have $f(i) = i+1$, so we want to find $\sum_{i=0}^{10^9} i+1$. In the third example, the frog can't make any jumps in any case. 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 Little Elephant from the Zoo of Lviv currently is on the military mission. There are N enemy buildings placed in a row and numbered from left to right strating from 0. Each building i (except the first and the last) has exactly two adjacent buildings with indices i-1 and i+1. The first and the last buildings have just a single adjacent building. Some of the buildings contain bombs. When bomb explodes in some building it destroys it and all adjacent to it buildings. You are given the string S of length N, where S_{i} is 1 if the i-th building contains bomb, 0 otherwise. Find for the Little Elephant the number of buildings that will not be destroyed after all bombs explode. Please note that all bombs explode simultaneously. ------ Input ------ The first line contains single integer T - the number of test cases. T test cases follow. The first line of each test case contains the single integer N - the number of buildings. The next line contains the string S of length N consisted only of digits 0 and 1. ------ Output ------ In T lines print T inetgers - the answers for the corresponding test cases. ------ Constraints ------ 1 ≤ T ≤ 100 1 ≤ N ≤ 1000 ----- Sample Input 1 ------ 3 3 010 5 10001 7 0000000 ----- Sample Output 1 ------ 0 1 7 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. Don Drumphet lives in a nice neighborhood, but one of his neighbors has started to let his house go. Don Drumphet wants to build a wall between his house and his neighbor’s, and is trying to get the neighborhood association to pay for it. He begins to solicit his neighbors to petition to get the association to build the wall. Unfortunately for Don Drumphet, he cannot read very well, has a very limited attention span, and can only remember two letters from each of his neighbors’ names. As he collects signatures, he insists that his neighbors keep truncating their names until two letters remain, and he can finally read them. Your code will show Full name of the neighbor and the truncated version of the name as an array. If the number of the characters in name is less than or equal to two, it will return an array containing only the name as is" 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 $n$ pairs of integers: $(a_1, b_1), (a_2, b_2), \dots , (a_n, b_n)$. This sequence is called bad if it is sorted in non-descending order by first elements or if it is sorted in non-descending order by second elements. Otherwise the sequence is good. There are examples of good and bad sequences: $s = [(1, 2), (3, 2), (3, 1)]$ is bad because the sequence of first elements is sorted: $[1, 3, 3]$; $s = [(1, 2), (3, 2), (1, 2)]$ is bad because the sequence of second elements is sorted: $[2, 2, 2]$; $s = [(1, 1), (2, 2), (3, 3)]$ is bad because both sequences (the sequence of first elements and the sequence of second elements) are sorted; $s = [(1, 3), (3, 3), (2, 2)]$ is good because neither the sequence of first elements $([1, 3, 2])$ nor the sequence of second elements $([3, 3, 2])$ is sorted. Calculate the number of permutations of size $n$ such that after applying this permutation to the sequence $s$ it turns into a good sequence. A permutation $p$ of size $n$ is a sequence $p_1, p_2, \dots , p_n$ consisting of $n$ distinct integers from $1$ to $n$ ($1 \le p_i \le n$). If you apply permutation $p_1, p_2, \dots , p_n$ to the sequence $s_1, s_2, \dots , s_n$ you get the sequence $s_{p_1}, s_{p_2}, \dots , s_{p_n}$. For example, if $s = [(1, 2), (1, 3), (2, 3)]$ and $p = [2, 3, 1]$ then $s$ turns into $[(1, 3), (2, 3), (1, 2)]$. -----Input----- The first line contains one integer $n$ ($1 \le n \le 3 \cdot 10^5$). The next $n$ lines contains description of sequence $s$. The $i$-th line contains two integers $a_i$ and $b_i$ ($1 \le a_i, b_i \le n$) — the first and second elements of $i$-th pair in the sequence. The sequence $s$ may contain equal elements. -----Output----- Print the number of permutations of size $n$ such that after applying this permutation to the sequence $s$ it turns into a good sequence. Print the answer modulo $998244353$ (a prime number). -----Examples----- Input 3 1 1 2 2 3 1 Output 3 Input 4 2 3 2 2 2 1 2 4 Output 0 Input 3 1 1 1 1 2 3 Output 4 -----Note----- In first test case there are six permutations of size $3$: if $p = [1, 2, 3]$, then $s = [(1, 1), (2, 2), (3, 1)]$ — bad sequence (sorted by first elements); if $p = [1, 3, 2]$, then $s = [(1, 1), (3, 1), (2, 2)]$ — bad sequence (sorted by second elements); if $p = [2, 1, 3]$, then $s = [(2, 2), (1, 1), (3, 1)]$ — good sequence; if $p = [2, 3, 1]$, then $s = [(2, 2), (3, 1), (1, 1)]$ — good sequence; if $p = [3, 1, 2]$, then $s = [(3, 1), (1, 1), (2, 2)]$ — bad sequence (sorted by second elements); if $p = [3, 2, 1]$, then $s = [(3, 1), (2, 2), (1, 1)]$ — good sequence. Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. You are given three positive integers x, y, n. Your task is to find the nearest fraction to fraction [Image] whose denominator is no more than n. Formally, you should find such pair of integers a, b (1 ≤ b ≤ n; 0 ≤ a) that the value $\|\frac{x}{y} - \frac{a}{b}\|$ is as minimal as possible. If there are multiple "nearest" fractions, choose the one with the minimum denominator. If there are multiple "nearest" fractions with the minimum denominator, choose the one with the minimum numerator. -----Input----- A single line contains three integers x, y, n (1 ≤ x, y, n ≤ 10^5). -----Output----- Print the required fraction in the format "a/b" (without quotes). -----Examples----- Input 3 7 6 Output 2/5 Input 7 2 4 Output 7/2 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 of the Insertion Sort algorithm which sorts a sequence A in ascending order. The algorithm should be based on the following pseudocode: for i = 1 to A.length-1 key = A[i] /* insert A[i] into the sorted sequence A[0,...,j-1] */ j = i - 1 while j >= 0 and A[j] > key A[j+1] = A[j] j-- A[j+1] = key Note that, indices for array elements are based on 0-origin. To illustrate the algorithms, your program should trace intermediate result for each step. Hint Template in C Constraints 1 ≤ N ≤ 100 Input The first line of the input includes an integer N, the number of elements in the sequence. In the second line, N elements of the sequence are given separated by a single space. Output The output consists of N lines. Please output the intermediate sequence in a line for each step. Elements of the sequence should be separated by single space. Examples Input 6 5 2 4 6 1 3 Output 5 2 4 6 1 3 2 5 4 6 1 3 2 4 5 6 1 3 2 4 5 6 1 3 1 2 4 5 6 3 1 2 3 4 5 6 Input 3 1 2 3 Output 1 2 3 1 2 3 1 2 3 Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Vasya has recently learned to type and log on to the Internet. He immediately entered a chat room and decided to say hello to everybody. Vasya typed the word s. It is considered that Vasya managed to say hello if several letters can be deleted from the typed word so that it resulted in the word "hello". For example, if Vasya types the word "ahhellllloou", it will be considered that he said hello, and if he types "hlelo", it will be considered that Vasya got misunderstood and he didn't manage to say hello. Determine whether Vasya managed to say hello by the given word s. Input The first and only line contains the word s, which Vasya typed. This word consisits of small Latin letters, its length is no less that 1 and no more than 100 letters. Output If Vasya managed to say hello, print "YES", otherwise print "NO". Examples Input ahhellllloou Output YES Input hlelo Output NO 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 decided to decorate his room because the New Year is soon. One of the main decorations that Polycarp will install is the garland he is going to solder himself. Simple garlands consisting of several lamps connected by one wire are too boring for Polycarp. He is going to solder a garland consisting of n lamps and n - 1 wires. Exactly one lamp will be connected to power grid, and power will be transmitted from it to other lamps by the wires. Each wire connectes exactly two lamps; one lamp is called the main lamp for this wire (the one that gets power from some other wire and transmits it to this wire), the other one is called the auxiliary lamp (the one that gets power from this wire). Obviously, each lamp has at most one wire that brings power to it (and this lamp is the auxiliary lamp for this wire, and the main lamp for all other wires connected directly to it). Each lamp has a brightness value associated with it, the i-th lamp has brightness 2^i. We define the importance of the wire as the sum of brightness values over all lamps that become disconnected from the grid if the wire is cut (and all other wires are still working). Polycarp has drawn the scheme of the garland he wants to make (the scheme depicts all n lamp and n - 1 wires, and the lamp that will be connected directly to the grid is marked; the wires are placed in such a way that the power can be transmitted to each lamp). After that, Polycarp calculated the importance of each wire, enumerated them from 1 to n - 1 in descending order of their importance, and then wrote the index of the main lamp for each wire (in the order from the first wire to the last one). The following day Polycarp bought all required components of the garland and decided to solder it — but he could not find the scheme. Fortunately, Polycarp found the list of indices of main lamps for all wires. Can you help him restore the original scheme? Input The first line contains one integer n (2 ≤ n ≤ 2 ⋅ 10^5) — the number of lamps. The second line contains n - 1 integers a_1, a_2, ..., a_{n - 1} (1 ≤ a_i ≤ n), where a_i is the index of the main lamp for the i-th wire (wires are numbered in descending order of importance). Output If it is impossible to restore the original scheme, print one integer -1. Otherwise print the scheme as follows. In the first line, print one integer k (1 ≤ k ≤ n) — the index of the lamp that is connected to the power grid. Then print n - 1 lines, each containing two integers x_i and y_i (1 ≤ x_i, y_i ≤ n, x_i ≠ y_i) — the indices of the lamps connected by some wire. The descriptions of the wires (and the lamps connected by a wire) can be printed in any order. The printed description must correspond to a scheme of a garland such that Polycarp could have written the list a_1, a_2, ..., a_{n - 1} from it. If there are multiple such schemes, output any of them. Example Input 6 3 6 3 1 5 Output 3 6 3 6 5 1 3 1 4 5 2 Note The scheme for the first example (R denotes the lamp connected to the grid, the numbers on wires are their importance values): <image> Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. One evening Rainbow Dash and Fluttershy have come up with a game. Since the ponies are friends, they have decided not to compete in the game but to pursue a common goal. The game starts on a square flat grid, which initially has the outline borders built up. Rainbow Dash and Fluttershy have flat square blocks with size $1\times1$, Rainbow Dash has an infinite amount of light blue blocks, Fluttershy has an infinite amount of yellow blocks. The blocks are placed according to the following rule: each newly placed block must touch the built on the previous turns figure by a side (note that the outline borders of the grid are built initially). At each turn, one pony can place any number of blocks of her color according to the game rules. Rainbow and Fluttershy have found out that they can build patterns on the grid of the game that way. They have decided to start with something simple, so they made up their mind to place the blocks to form a chess coloring. Rainbow Dash is well-known for her speed, so she is interested in the minimum number of turns she and Fluttershy need to do to get a chess coloring, covering the whole grid with blocks. Please help her find that number! Since the ponies can play many times on different boards, Rainbow Dash asks you to find the minimum numbers of turns for several grids of the games. The chess coloring in two colors is the one in which each square is neighbor by side only with squares of different colors. -----Input----- The first line contains a single integer $T$ ($1 \le T \le 100$): the number of grids of the games. Each of the next $T$ lines contains a single integer $n$ ($1 \le n \le 10^9$): the size of the side of the grid of the game. -----Output----- For each grid of the game print the minimum number of turns required to build a chess coloring pattern out of blocks on it. -----Example----- Input 2 3 4 Output 2 3 -----Note----- For $3\times3$ grid ponies can make two following moves: [Image] 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 window of Takahashi's room has a width of A. There are two curtains hung over the window, each of which has a horizontal length of B. (Vertically, the curtains are long enough to cover the whole window.) We will close the window so as to minimize the total horizontal length of the uncovered part of the window. Find the total horizontal length of the uncovered parts of the window then. -----Constraints----- - 1 \leq A \leq 100 - 1 \leq B \leq 100 - A and B are integers. -----Input----- Input is given from Standard Input in the following format: A B -----Output----- Print the total horizontal length of the uncovered parts of the window. -----Sample Input----- 12 4 -----Sample Output----- 4 We have a window with a horizontal length of 12, and two curtains, each of length 4, that cover both ends of the window, for example. The uncovered part has a horizontal length of 4. 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 two bottles for holding water. Bottle 1 can hold up to A milliliters of water, and now it contains B milliliters of water. Bottle 2 contains C milliliters of water. We will transfer water from Bottle 2 to Bottle 1 as much as possible. How much amount of water will remain in Bottle 2? -----Constraints----- - All values in input are integers. - 1 \leq B \leq A \leq 20 - 1 \leq C \leq 20 -----Input----- Input is given from Standard Input in the following format: A B C -----Output----- Print the integer representing the amount of water, in milliliters, that will remain in Bottle 2. -----Sample Input----- 6 4 3 -----Sample Output----- 1 We will transfer two milliliters of water from Bottle 2 to Bottle 1, and one milliliter of water will remain in Bottle 2. 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. Vladik had started reading a complicated book about algorithms containing n pages. To improve understanding of what is written, his friends advised him to read pages in some order given by permutation P = [p_1, p_2, ..., p_{n}], where p_{i} denotes the number of page that should be read i-th in turn. Sometimes Vladik’s mom sorted some subsegment of permutation P from position l to position r inclusive, because she loves the order. For every of such sorting Vladik knows number x — what index of page in permutation he should read. He is wondered if the page, which he will read after sorting, has changed. In other words, has p_{x} changed? After every sorting Vladik return permutation to initial state, so you can assume that each sorting is independent from each other. -----Input----- First line contains two space-separated integers n, m (1 ≤ n, m ≤ 10^4) — length of permutation and number of times Vladik's mom sorted some subsegment of the book. Second line contains n space-separated integers p_1, p_2, ..., p_{n} (1 ≤ p_{i} ≤ n) — permutation P. Note that elements in permutation are distinct. Each of the next m lines contains three space-separated integers l_{i}, r_{i}, x_{i} (1 ≤ l_{i} ≤ x_{i} ≤ r_{i} ≤ n) — left and right borders of sorted subsegment in i-th sorting and position that is interesting to Vladik. -----Output----- For each mom’s sorting on it’s own line print "Yes", if page which is interesting to Vladik hasn't changed, or "No" otherwise. -----Examples----- Input 5 5 5 4 3 2 1 1 5 3 1 3 1 2 4 3 4 4 4 2 5 3 Output Yes No Yes Yes No Input 6 5 1 4 3 2 5 6 2 4 3 1 6 2 4 5 4 1 3 3 2 6 3 Output Yes No Yes No Yes -----Note----- Explanation of first test case: [1, 2, 3, 4, 5] — permutation after sorting, 3-rd element hasn’t changed, so answer is "Yes". [3, 4, 5, 2, 1] — permutation after sorting, 1-st element has changed, so answer is "No". [5, 2, 3, 4, 1] — permutation after sorting, 3-rd element hasn’t changed, so answer is "Yes". [5, 4, 3, 2, 1] — permutation after sorting, 4-th element hasn’t changed, so answer is "Yes". [5, 1, 2, 3, 4] — permutation after sorting, 3-rd element has changed, so answer is "No". 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. ## Objective Given a number `n` we will define it's sXORe to be `0 XOR 1 XOR 2 ... XOR n` where `XOR` is the [bitwise XOR operator](https://en.wikipedia.org/wiki/Bitwise_operation#XOR). Write a function that takes `n` and returns it's sXORe. ## Examples \| n \| sXORe n \|---------\|-------- \| 0 \| 0 \| 1 \| 1 \| 50 \| 51 \| 1000000 \| 1000000 --- 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. Consider a table of size $n \times m$, initially fully white. Rows are numbered $1$ through $n$ from top to bottom, columns $1$ through $m$ from left to right. Some square inside the table with odd side length was painted black. Find the center of this square. -----Input----- The first line contains two integers $n$ and $m$ ($1 \le n, m \le 115$) — the number of rows and the number of columns in the table. The $i$-th of the next $n$ lines contains a string of $m$ characters $s_{i1} s_{i2} \ldots s_{im}$ ($s_{ij}$ is 'W' for white cells and 'B' for black cells), describing the $i$-th row of the table. -----Output----- Output two integers $r$ and $c$ ($1 \le r \le n$, $1 \le c \le m$) separated by a space — the row and column numbers of the center of the black square. -----Examples----- Input 5 6 WWBBBW WWBBBW WWBBBW WWWWWW WWWWWW Output 2 4 Input 3 3 WWW BWW WWW Output 2 1 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. Finish the solution so that it takes an input `n` (integer) and returns a string that is the decimal representation of the number grouped by commas after every 3 digits. Assume: `0 <= n < 2147483647` ## Examples ``` 1 -> "1" 10 -> "10" 100 -> "100" 1000 -> "1,000" 10000 -> "10,000" 100000 -> "100,000" 1000000 -> "1,000,000" 35235235 -> "35,235,235" ``` 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. Ivan is a student at Berland State University (BSU). There are n days in Berland week, and each of these days Ivan might have some classes at the university. There are m working hours during each Berland day, and each lesson at the university lasts exactly one hour. If at some day Ivan's first lesson is during i-th hour, and last lesson is during j-th hour, then he spends j - i + 1 hours in the university during this day. If there are no lessons during some day, then Ivan stays at home and therefore spends 0 hours in the university. Ivan doesn't like to spend a lot of time in the university, so he has decided to skip some lessons. He cannot skip more than k lessons during the week. After deciding which lessons he should skip and which he should attend, every day Ivan will enter the university right before the start of the first lesson he does not skip, and leave it after the end of the last lesson he decides to attend. If Ivan skips all lessons during some day, he doesn't go to the university that day at all. Given n, m, k and Ivan's timetable, can you determine the minimum number of hours he has to spend in the university during one week, if he cannot skip more than k lessons? -----Input----- The first line contains three integers n, m and k (1 ≤ n, m ≤ 500, 0 ≤ k ≤ 500) — the number of days in the Berland week, the number of working hours during each day, and the number of lessons Ivan can skip, respectively. Then n lines follow, i-th line containing a binary string of m characters. If j-th character in i-th line is 1, then Ivan has a lesson on i-th day during j-th hour (if it is 0, there is no such lesson). -----Output----- Print the minimum number of hours Ivan has to spend in the university during the week if he skips not more than k lessons. -----Examples----- Input 2 5 1 01001 10110 Output 5 Input 2 5 0 01001 10110 Output 8 -----Note----- In the first example Ivan can skip any of two lessons during the first day, so he spends 1 hour during the first day and 4 hours during the second day. In the second example Ivan can't skip any lessons, so he spends 4 hours every day. 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 the big birthday party, Katie still wanted Shiro to have some more fun. Later, she came up with a game called treasure hunt. Of course, she invited her best friends Kuro and Shiro to play with her. The three friends are very smart so they passed all the challenges very quickly and finally reached the destination. But the treasure can only belong to one cat so they started to think of something which can determine who is worthy of the treasure. Instantly, Kuro came up with some ribbons. A random colorful ribbon is given to each of the cats. Each color of the ribbon can be represented as an uppercase or lowercase Latin letter. Let's call a consecutive subsequence of colors that appears in the ribbon a subribbon. The beauty of a ribbon is defined as the maximum number of times one of its subribbon appears in the ribbon. The more the subribbon appears, the more beautiful is the ribbon. For example, the ribbon aaaaaaa has the beauty of $7$ because its subribbon a appears $7$ times, and the ribbon abcdabc has the beauty of $2$ because its subribbon abc appears twice. The rules are simple. The game will have $n$ turns. Every turn, each of the cats must change strictly one color (at one position) in his/her ribbon to an arbitrary color which is different from the unchanged one. For example, a ribbon aaab can be changed into acab in one turn. The one having the most beautiful ribbon after $n$ turns wins the treasure. Could you find out who is going to be the winner if they all play optimally? -----Input----- The first line contains an integer $n$ ($0 \leq n \leq 10^{9}$) — the number of turns. Next 3 lines contain 3 ribbons of Kuro, Shiro and Katie one per line, respectively. Each ribbon is a string which contains no more than $10^{5}$ uppercase and lowercase Latin letters and is not empty. It is guaranteed that the length of all ribbons are equal for the purpose of fairness. Note that uppercase and lowercase letters are considered different colors. -----Output----- Print the name of the winner ("Kuro", "Shiro" or "Katie"). If there are at least two cats that share the maximum beauty, print "Draw". -----Examples----- Input 3 Kuroo Shiro Katie Output Kuro Input 7 treasurehunt threefriends hiCodeforces Output Shiro Input 1 abcabc cbabac ababca Output Katie Input 15 foPaErcvJ mZaxowpbt mkuOlaHRE Output Draw -----Note----- In the first example, after $3$ turns, Kuro can change his ribbon into ooooo, which has the beauty of $5$, while reaching such beauty for Shiro and Katie is impossible (both Shiro and Katie can reach the beauty of at most $4$, for example by changing Shiro's ribbon into SSiSS and changing Katie's ribbon into Kaaaa). Therefore, the winner is Kuro. In the fourth example, since the length of each of the string is $9$ and the number of turn is $15$, everyone can change their ribbons in some way to reach the maximal beauty of $9$ by changing their strings into zzzzzzzzz after 9 turns, and repeatedly change their strings into azzzzzzzz and then into zzzzzzzzz thrice. Therefore, the game ends in a draw. 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 an administrator of a public page of organization "Mouse and keyboard" and his everyday duty is to publish news from the world of competitive programming. For each news he also creates a list of hashtags to make searching for a particular topic more comfortable. For the purpose of this problem we define hashtag as a string consisting of lowercase English letters and exactly one symbol '#' located at the beginning of the string. The length of the hashtag is defined as the number of symbols in it without the symbol '#'. The head administrator of the page told Vasya that hashtags should go in lexicographical order (take a look at the notes section for the definition). Vasya is lazy so he doesn't want to actually change the order of hashtags in already published news. Instead, he decided to delete some suffixes (consecutive characters at the end of the string) of some of the hashtags. He is allowed to delete any number of characters, even the whole string except for the symbol '#'. Vasya wants to pick such a way to delete suffixes that the total number of deleted symbols is minimum possible. If there are several optimal solutions, he is fine with any of them. -----Input----- The first line of the input contains a single integer n (1 ≤ n ≤ 500 000) — the number of hashtags being edited now. Each of the next n lines contains exactly one hashtag of positive length. It is guaranteed that the total length of all hashtags (i.e. the total length of the string except for characters '#') won't exceed 500 000. -----Output----- Print the resulting hashtags in any of the optimal solutions. -----Examples----- Input 3 #book #bigtown #big Output #b #big #big Input 3 #book #cool #cold Output #book #co #cold Input 4 #car #cart #art #at Output # # #art #at Input 3 #apple #apple #fruit Output #apple #apple #fruit -----Note----- Word a_1, a_2, ..., a_{m} of length m is lexicographically not greater than word b_1, b_2, ..., b_{k} of length k, if one of two conditions hold: at first position i, such that a_{i} ≠ b_{i}, the character a_{i} goes earlier in the alphabet than character b_{i}, i.e. a has smaller character than b in the first position where they differ; if there is no such position i and m ≤ k, i.e. the first word is a prefix of the second or two words are equal. The sequence of words is said to be sorted in lexicographical order if each word (except the last one) is lexicographically not greater than the next word. For the words consisting of lowercase English letters the lexicographical order coincides with the alphabet word order in the dictionary. According to the above definition, if a hashtag consisting of one character '#' it is lexicographically not greater than any other valid hashtag. That's why in the third sample we can't keep first two hashtags unchanged and shorten the other two. 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 once wrote a sad love song and shared it to Vasya. The song is a string consisting of lowercase English letters. Vasya made up $q$ questions about this song. Each question is about a subsegment of the song starting from the $l$-th letter to the $r$-th letter. Vasya considers a substring made up from characters on this segment and repeats each letter in the subsegment $k$ times, where $k$ is the index of the corresponding letter in the alphabet. For example, if the question is about the substring "abbcb", then Vasya repeats letter 'a' once, each of the letters 'b' twice, letter 'c" three times, so that the resulting string is "abbbbcccbb", its length is $10$. Vasya is interested about the length of the resulting string. Help Petya find the length of each string obtained by Vasya. -----Input----- The first line contains two integers $n$ and $q$ ($1\leq n\leq 100000$, $1\leq q \leq 100000$) — the length of the song and the number of questions. The second line contains one string $s$ — the song, consisting of $n$ lowercase letters of English letters. Vasya's questions are contained in the next $q$ lines. Each line contains two integers $l$ and $r$ ($1 \leq l \leq r \leq n$) — the bounds of the question. -----Output----- Print $q$ lines: for each question print the length of the string obtained by Vasya. -----Examples----- Input 7 3 abacaba 1 3 2 5 1 7 Output 4 7 11 Input 7 4 abbabaa 1 3 5 7 6 6 2 4 Output 5 4 1 5 Input 13 7 sonoshikumiwo 1 5 2 10 7 7 1 13 4 8 2 5 3 9 Output 82 125 9 191 62 63 97 -----Note----- In the first example Vasya is interested in three questions. In the first question Vasya considers the substring "aba", that transforms to "abba", so the answer is equal to $4$. In the second question Vasya considers "baca", that transforms to "bbaccca", so the answer is $7$. In the third question Vasya considers the string "abacaba",that transforms to "abbacccabba" of length $11$. 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 of length N, a_1,a_2,...,a_N. For each 1≤i≤N, you have three choices: add 1 to a_i, subtract 1 from a_i or do nothing. After these operations, you select an integer X and count the number of i such that a_i=X. Maximize this count by making optimal choices. Constraints * 1≤N≤10^5 * 0≤a_i<10^5 (1≤i≤N) * a_i is an integer. Input The input is given from Standard Input in the following format: N a_1 a_2 .. a_N Output Print the maximum possible number of i such that a_i=X. Examples Input 7 3 1 4 1 5 9 2 Output 4 Input 10 0 1 2 3 4 5 6 7 8 9 Output 3 Input 1 99999 Output 1 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 Berland capital is shaken with three bold crimes committed by the Pihsters, a notorious criminal gang. The Berland capital's map is represented by an n × m rectangular table. Each cell of the table on the map represents some districts of the capital. The capital's main detective Polycarpus took a map and marked there the districts where the first three robberies had been committed as asterisks. Deduction tells Polycarpus that the fourth robbery will be committed in such district, that all four robbed districts will form the vertices of some rectangle, parallel to the sides of the map. Polycarpus is good at deduction but he's hopeless at math. So he asked you to find the district where the fourth robbery will be committed. Input The first line contains two space-separated integers n and m (2 ≤ n, m ≤ 100) — the number of rows and columns in the table, correspondingly. Each of the next n lines contains m characters — the description of the capital's map. Each character can either be a "." (dot), or an "" (asterisk). A character equals "" if the corresponding district has been robbed. Otherwise, it equals ".". It is guaranteed that the map has exactly three characters "" and we can always find the fourth district that meets the problem requirements. Output Print two integers — the number of the row and the number of the column of the city district that is the fourth one to be robbed. The rows are numbered starting from one from top to bottom and the columns are numbered starting from one from left to right. Examples Input 3 2 . .. ** Output 1 1 Input 3 3 . *.. ... Output 2 3 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 takes part in the orienteering competition. There are n checkpoints located along the line at coordinates x_1, x_2, ..., x_{n}. Vasya starts at the point with coordinate a. His goal is to visit at least n - 1 checkpoint in order to finish the competition. Participant are allowed to visit checkpoints in arbitrary order. Vasya wants to pick such checkpoints and the order of visiting them that the total distance travelled is minimized. He asks you to calculate this minimum possible value. -----Input----- The first line of the input contains two integers n and a (1 ≤ n ≤ 100 000, - 1 000 000 ≤ a ≤ 1 000 000) — the number of checkpoints and Vasya's starting position respectively. The second line contains n integers x_1, x_2, ..., x_{n} ( - 1 000 000 ≤ x_{i} ≤ 1 000 000) — coordinates of the checkpoints. -----Output----- Print one integer — the minimum distance Vasya has to travel in order to visit at least n - 1 checkpoint. -----Examples----- Input 3 10 1 7 12 Output 7 Input 2 0 11 -10 Output 10 Input 5 0 0 0 1000 0 0 Output 0 -----Note----- In the first sample Vasya has to visit at least two checkpoints. The optimal way to achieve this is the walk to the third checkpoints (distance is 12 - 10 = 2) and then proceed to the second one (distance is 12 - 7 = 5). The total distance is equal to 2 + 5 = 7. In the second sample it's enough to visit only one checkpoint so Vasya should just walk to the point - 10. 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. Based on some criterion, Snuke divided the integers from 1 through 12 into three groups as shown in the figure below. Given two integers x and y (1 ≤ x < y ≤ 12), determine whether they belong to the same group. -----Constraints----- - x and y are integers. - 1 ≤ x < y ≤ 12 -----Input----- Input is given from Standard Input in the following format: x y -----Output----- If x and y belong to the same group, print Yes; otherwise, print No. -----Sample Input----- 1 3 -----Sample Output----- Yes Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. N Snukes called Snuke 1, Snuke 2, ..., Snuke N live in a town. There are K kinds of snacks sold in this town, called Snack 1, Snack 2, ..., Snack K. The following d_i Snukes have Snack i: Snuke A_{i, 1}, A_{i, 2}, \cdots, A_{i, {d_i}}. Takahashi will walk around this town and make mischief on the Snukes who have no snacks. How many Snukes will fall victim to Takahashi's mischief? -----Constraints----- - All values in input are integers. - 1 \leq N \leq 100 - 1 \leq K \leq 100 - 1 \leq d_i \leq N - 1 \leq A_{i, 1} < \cdots < A_{i, d_i} \leq N -----Input----- Input is given from Standard Input in the following format: N K d_1 A_{1, 1} \cdots A_{1, d_1} \vdots d_K A_{K, 1} \cdots A_{K, d_K} -----Output----- Print the answer. -----Sample Input----- 3 2 2 1 3 1 3 -----Sample Output----- 1 - Snuke 1 has Snack 1. - Snuke 2 has no snacks. - Snuke 3 has Snack 1 and 2. Thus, there will be one victim: Snuke 2. 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. Kuro is living in a country called Uberland, consisting of $n$ towns, numbered from $1$ to $n$, and $n - 1$ bidirectional roads connecting these towns. It is possible to reach each town from any other. Each road connects two towns $a$ and $b$. Kuro loves walking and he is planning to take a walking marathon, in which he will choose a pair of towns $(u, v)$ ($u \neq v$) and walk from $u$ using the shortest path to $v$ (note that $(u, v)$ is considered to be different from $(v, u)$). Oddly, there are 2 special towns in Uberland named Flowrisa (denoted with the index $x$) and Beetopia (denoted with the index $y$). Flowrisa is a town where there are many strong-scent flowers, and Beetopia is another town where many bees live. In particular, Kuro will avoid any pair of towns $(u, v)$ if on the path from $u$ to $v$, he reaches Beetopia after he reached Flowrisa, since the bees will be attracted with the flower smell on Kuro’s body and sting him. Kuro wants to know how many pair of city $(u, v)$ he can take as his route. Since he’s not really bright, he asked you to help him with this problem. -----Input----- The first line contains three integers $n$, $x$ and $y$ ($1 \leq n \leq 3 \cdot 10^5$, $1 \leq x, y \leq n$, $x \ne y$) - the number of towns, index of the town Flowrisa and index of the town Beetopia, respectively. $n - 1$ lines follow, each line contains two integers $a$ and $b$ ($1 \leq a, b \leq n$, $a \ne b$), describes a road connecting two towns $a$ and $b$. It is guaranteed that from each town, we can reach every other town in the city using the given roads. That is, the given map of towns and roads is a tree. -----Output----- A single integer resembles the number of pair of towns $(u, v)$ that Kuro can use as his walking route. -----Examples----- Input 3 1 3 1 2 2 3 Output 5 Input 3 1 3 1 2 1 3 Output 4 -----Note----- On the first example, Kuro can choose these pairs: $(1, 2)$: his route would be $1 \rightarrow 2$, $(2, 3)$: his route would be $2 \rightarrow 3$, $(3, 2)$: his route would be $3 \rightarrow 2$, $(2, 1)$: his route would be $2 \rightarrow 1$, $(3, 1)$: his route would be $3 \rightarrow 2 \rightarrow 1$. Kuro can't choose pair $(1, 3)$ since his walking route would be $1 \rightarrow 2 \rightarrow 3$, in which Kuro visits town $1$ (Flowrisa) and then visits town $3$ (Beetopia), which is not allowed (note that pair $(3, 1)$ is still allowed because although Kuro visited Flowrisa and Beetopia, he did not visit them in that order). On the second example, Kuro can choose the following pairs: $(1, 2)$: his route would be $1 \rightarrow 2$, $(2, 1)$: his route would be $2 \rightarrow 1$, $(3, 2)$: his route would be $3 \rightarrow 1 \rightarrow 2$, $(3, 1)$: his route would be $3 \rightarrow 1$. 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. Paul is at the orchestra. The string section is arranged in an r × c rectangular grid and is filled with violinists with the exception of n violists. Paul really likes violas, so he would like to take a picture including at least k of them. Paul can take a picture of any axis-parallel rectangle in the orchestra. Count the number of possible pictures that Paul can take. Two pictures are considered to be different if the coordinates of corresponding rectangles are different. -----Input----- The first line of input contains four space-separated integers r, c, n, k (1 ≤ r, c, n ≤ 10, 1 ≤ k ≤ n) — the number of rows and columns of the string section, the total number of violas, and the minimum number of violas Paul would like in his photograph, respectively. The next n lines each contain two integers x_{i} and y_{i} (1 ≤ x_{i} ≤ r, 1 ≤ y_{i} ≤ c): the position of the i-th viola. It is guaranteed that no location appears more than once in the input. -----Output----- Print a single integer — the number of photographs Paul can take which include at least k violas. -----Examples----- Input 2 2 1 1 1 2 Output 4 Input 3 2 3 3 1 1 3 1 2 2 Output 1 Input 3 2 3 2 1 1 3 1 2 2 Output 4 -----Note----- We will use '' to denote violinists and '#' to denote violists. In the first sample, the orchestra looks as follows # ** Paul can take a photograph of just the viola, the 1 × 2 column containing the viola, the 2 × 1 row containing the viola, or the entire string section, for 4 pictures total. In the second sample, the orchestra looks as follows #* # # Paul must take a photograph of the entire section. In the third sample, the orchestra looks the same as in the second sample. 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 will receive 3 points for solving this problem. Manao is designing the genetic code for a new type of algae to efficiently produce fuel. Specifically, Manao is focusing on a stretch of DNA that encodes one protein. The stretch of DNA is represented by a string containing only the characters 'A', 'T', 'G' and 'C'. Manao has determined that if the stretch of DNA contains a maximal sequence of consecutive identical nucleotides that is of even length, then the protein will be nonfunctional. For example, consider a protein described by DNA string "GTTAAAG". It contains four maximal sequences of consecutive identical nucleotides: "G", "TT", "AAA", and "G". The protein is nonfunctional because sequence "TT" has even length. Manao is trying to obtain a functional protein from the protein he currently has. Manao can insert additional nucleotides into the DNA stretch. Each additional nucleotide is a character from the set {'A', 'T', 'G', 'C'}. Manao wants to determine the minimum number of insertions necessary to make the DNA encode a functional protein. -----Input----- The input consists of a single line, containing a string s of length n (1 ≤ n ≤ 100). Each character of s will be from the set {'A', 'T', 'G', 'C'}. This problem doesn't have subproblems. You will get 3 points for the correct submission. -----Output----- The program should print on one line a single integer representing the minimum number of 'A', 'T', 'G', 'C' characters that are required to be inserted into the input string in order to make all runs of identical characters have odd length. -----Examples----- Input GTTAAAG Output 1 Input AACCAACCAAAAC Output 5 -----Note----- In the first example, it is sufficient to insert a single nucleotide of any type between the two 'T's in the sequence to restore the functionality of the protein. 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 a programmer on the development team for new recording media. This recording medium can be randomly accessed for reading and erasing data. On the other hand, when writing data, access is always performed in order from the beginning, and only the first free space found can be written. You have begun to build a file system for this recording medium. In this file system, data is written in order from the first free area due to the limitation of the recording medium. If another data exists in the middle of writing, the remaining data is written from the free area behind it. Data is written in units called sectors. Sectors are assigned numbers starting from 0, which indicate their physical location on the storage medium. The sector numbers are assigned as 0, 1, 2, 3, ... In order from the beginning to the back of the storage medium. There are three commands in the file system: write, delete, and sector reference. Your job is to reproduce the behavior of this file system and then write a program that outputs which files are located in the target sector when the reference command is executed. In the initial state, nothing is written on the recording medium. For example, let's look at the first example of Sample Input. The first instruction writes a file with an identifier of 0 and a size of 2. In the initial state, nothing is written to the recording medium, that is, all sectors are free areas, so writing is performed to the first two sectors, that is, the 0th sector and the 1st sector. Therefore, the storage medium after writing is as follows. 0 0 Sky Sky Sky Sky Sky Sky ... The second instruction writes a file with the identifier 1 to the second and third sectors. The state of the storage medium after this is as follows. 0 0 1 1 Sky Sky Sky Sky ... The third instruction deletes the file with the identifier 0. The state of the storage medium is as follows. Sky Sky 1 1 Sky Sky Sky Sky ... The fourth instruction writes the file with the identifier 2 to the 0th, 1st, 4th, and 5th sectors. 2 2 1 1 2 2 Sky Sky ... The last instruction refers to the third sector. Now that you have a file with the identifier 1 in the third sector, your program should output 1. Input The input consists of multiple datasets. Each dataset is given in the following format. > N > Command1 > Command2 > ... > CommandN N is the number of commands executed (1 ≤ N ≤ 10,000), and Commandi is the i-th command to be executed. Each command consists of a command name and one or two arguments. The command name consists of only one letter and can be either "W", "D", or "R". There is a space between the command and the argument, and between the argument and the argument. "W" represents a write command. Two arguments I (0 ≤ I ≤ 109) and S (1 ≤ S ≤ 109) are given. Each shows the identifier of the file to write and the number of sectors required to store the file. "D" represents the delete command. One argument I (0 ≤ I ≤ 109) is given. Indicates the identifier of the file to be deleted. "R" represents the reference command. One argument P (0 ≤ P ≤ 109) is given. Indicates the number of the sector to be referenced. It can be assumed that sectors with numbers higher than 109 do not need to be accessed. Also, it is guaranteed that a file with the same file identifier will not be written multiple times. The end of the input is indicated by one line containing one 0. Output For each dataset, output the identifier of the file referenced by that command on a single line each time a referenced command appears. If no file has been written to the referenced sector, output -1 instead of the file identifier. Put a blank line after each dataset. Sample Input 6 W 0 2 W 1 2 D 0 W 2 4 R 3 R 1 1 R 1000000000 0 Output for the Sample Input 1 2 -1 Example Input 6 W 0 2 W 1 2 D 0 W 2 4 R 3 R 1 1 R 1000000000 0 Output 1 2 -1 Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Pasha has two hamsters: Arthur and Alexander. Pasha put n apples in front of them. Pasha knows which apples Arthur likes. Similarly, Pasha knows which apples Alexander likes. Pasha doesn't want any conflict between the hamsters (as they may like the same apple), so he decided to distribute the apples between the hamsters on his own. He is going to give some apples to Arthur and some apples to Alexander. It doesn't matter how many apples each hamster gets but it is important that each hamster gets only the apples he likes. It is possible that somebody doesn't get any apples. Help Pasha distribute all the apples between the hamsters. Note that Pasha wants to distribute all the apples, not just some of them. -----Input----- The first line contains integers n, a, b (1 ≤ n ≤ 100; 1 ≤ a, b ≤ n) — the number of apples Pasha has, the number of apples Arthur likes and the number of apples Alexander likes, correspondingly. The next line contains a distinct integers — the numbers of the apples Arthur likes. The next line contains b distinct integers — the numbers of the apples Alexander likes. Assume that the apples are numbered from 1 to n. The input is such that the answer exists. -----Output----- Print n characters, each of them equals either 1 or 2. If the i-h character equals 1, then the i-th apple should be given to Arthur, otherwise it should be given to Alexander. If there are multiple correct answers, you are allowed to print any of them. -----Examples----- Input 4 2 3 1 2 2 3 4 Output 1 1 2 2 Input 5 5 2 3 4 1 2 5 2 3 Output 1 1 1 1 1 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 number "zero" is called "love" (or "l'oeuf" to be precise, literally means "egg" in French), for example when denoting the zero score in a game of tennis. Aki is fond of numbers, especially those with trailing zeros. For example, the number $9200$ has two trailing zeros. Aki thinks the more trailing zero digits a number has, the prettier it is. However, Aki believes, that the number of trailing zeros of a number is not static, but depends on the base (radix) it is represented in. Thus, he considers a few scenarios with some numbers and bases. And now, since the numbers he used become quite bizarre, he asks you to help him to calculate the beauty of these numbers. Given two integers $n$ and $b$ (in decimal notation), your task is to calculate the number of trailing zero digits in the $b$-ary (in the base/radix of $b$) representation of $n\,!$ (factorial of $n$). -----Input----- The only line of the input contains two integers $n$ and $b$ ($1 \le n \le 10^{18}$, $2 \le b \le 10^{12}$). -----Output----- Print an only integer — the number of trailing zero digits in the $b$-ary representation of $n!$ -----Examples----- Input 6 9 Output 1 Input 38 11 Output 3 Input 5 2 Output 3 Input 5 10 Output 1 -----Note----- In the first example, $6!_{(10)} = 720_{(10)} = 880_{(9)}$. In the third and fourth example, $5!_{(10)} = 120_{(10)} = 1111000_{(2)}$. The representation of the number $x$ in the $b$-ary base is $d_1, d_2, \ldots, d_k$ if $x = d_1 b^{k - 1} + d_2 b^{k - 2} + \ldots + d_k b^0$, where $d_i$ are integers and $0 \le d_i \le b - 1$. For example, the number $720$ from the first example is represented as $880_{(9)}$ since $720 = 8 \cdot 9^2 + 8 \cdot 9 + 0 \cdot 1$. You can read more about bases here. 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. Go around the Labyrinth Explorer Taro got a floor plan of a labyrinth. The floor of this labyrinth is in the form of a two-dimensional grid. Each of the cells on the floor plan corresponds to a room and is indicated whether it can be entered or not. The labyrinth has only one entrance located at the northwest corner, which is upper left on the floor plan. There is a treasure chest in each of the rooms at other three corners, southwest, southeast, and northeast. To get a treasure chest, Taro must move onto the room where the treasure chest is placed. Taro starts from the entrance room and repeats moving to one of the enterable adjacent rooms in the four directions, north, south, east, or west. He wants to collect all the three treasure chests and come back to the entrance room. A bad news for Taro is that, the labyrinth is quite dilapidated and even for its enterable rooms except for the entrance room, floors are so fragile that, once passed over, it will collapse and the room becomes not enterable. Determine whether it is possible to collect all the treasure chests and return to the entrance. Input The input consists of at most 100 datasets, each in the following format. N M c1,1...c1,M ... cN,1...cN,M The first line contains N, which is the number of rooms in the north-south direction, and M, which is the number of rooms in the east-west direction. N and M are integers satisfying 2 ≤ N ≤ 50 and 2 ≤ M ≤ 50. Each of the following N lines contains a string of length M. The j-th character ci,j of the i-th string represents the state of the room (i,j), i-th from the northernmost and j-th from the westernmost; the character is the period ('`.`') if the room is enterable and the number sign ('`#`') if the room is not enterable. The entrance is located at (1,1), and the treasure chests are placed at (N,1), (N,M) and (1,M). All of these four rooms are enterable. Taro cannot go outside the given N × M rooms. The end of the input is indicated by a line containing two zeros. Output For each dataset, output `YES` if it is possible to collect all the treasure chests and return to the entrance room, and otherwise, output `NO` in a single line. Sample Input 3 3 ... .#. ... 5 5 ..#.. ..... .... ..... ..... 3 8 ..#..... ........ .....#.. 3 5 ..#.. ..... ..#.. 4 4 .... .... ..## ..#. 0 0 Output for the Sample Input YES NO YES NO NO Example Input 3 3 ... .#. ... 5 5 ..#.. ..... #.... ..... ..... 3 8 ..#..... ........ .....#.. 3 5 ..#.. ..... ..#.. 4 4 .... .... ..## ..#. 0 0 Output YES NO YES NO NO 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 number of chocolate pieces were prepared for a training camp. The camp had N participants and lasted for D days. The i-th participant (1 \leq i \leq N) ate one chocolate piece on each of the following days in the camp: the 1-st day, the (A_i + 1)-th day, the (2A_i + 1)-th day, and so on. As a result, there were X chocolate pieces remaining at the end of the camp. During the camp, nobody except the participants ate chocolate pieces. Find the number of chocolate pieces prepared at the beginning of the camp. -----Constraints----- - 1 \leq N \leq 100 - 1 \leq D \leq 100 - 1 \leq X \leq 100 - 1 \leq A_i \leq 100 (1 \leq i \leq N) - All input values are integers. -----Input----- Input is given from Standard Input in the following format: N D X A_1 A_2 : A_N -----Output----- Find the number of chocolate pieces prepared at the beginning of the camp. -----Sample Input----- 3 7 1 2 5 10 -----Sample Output----- 8 The camp has 3 participants and lasts for 7 days. Each participant eats chocolate pieces as follows: - The first participant eats one chocolate piece on Day 1, 3, 5 and 7, for a total of four. - The second participant eats one chocolate piece on Day 1 and 6, for a total of two. - The third participant eats one chocolate piece only on Day 1, for a total of one. Since the number of pieces remaining at the end of the camp is one, the number of pieces prepared at the beginning of the camp is 1 + 4 + 2 + 1 = 8. 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 is a version of problem D from the same contest with some additional constraints and tasks. There are $n$ candies in a candy box. The type of the $i$-th candy is $a_i$ ($1 \le a_i \le n$). You have to prepare a gift using some of these candies with the following restriction: the numbers of candies of each type presented in a gift should be all distinct (i. e. for example, a gift having two candies of type $1$ and two candies of type $2$ is bad). It is possible that multiple types of candies are completely absent from the gift. It is also possible that not all candies of some types will be taken to a gift. You really like some of the candies and don't want to include them into the gift, but you want to eat them yourself instead. For each candy, a number $f_i$ is given, which is equal to $0$ if you really want to keep $i$-th candy for yourself, or $1$ if you don't mind including it into your gift. It is possible that two candies of the same type have different values of $f_i$. You want your gift to be as large as possible, but you don't want to include too many of the candies you want to eat into the gift. So, you want to calculate the maximum possible number of candies that can be included into a gift, and among all ways to choose maximum number of candies, you want to maximize the number of candies having $f_i = 1$ in your gift. You have to answer $q$ independent queries. If you are Python programmer, consider using PyPy instead of Python when you submit your code. -----Input----- The first line of the input contains one integer $q$ ($1 \le q \le 2 \cdot 10^5$) — the number of queries. The first line of each query contains one integer $n$ ($1 \le n \le 2 \cdot 10^5$) — the number of candies. Then $n$ lines follow, each containing two integers $a_i$ and $f_i$ ($1 \le a_i \le n$, $0 \le f_i \le 1$), where $a_i$ is the type of the $i$-th candy, and $f_i$ denotes whether you want to keep the $i$-th candy for yourself ($0$ if you want to keep it, $1$ if you don't mind giving it away). It is guaranteed that the sum of $n$ over all queries does not exceed $2 \cdot 10^5$. -----Output----- For each query print two integers: the maximum number of candies in a gift you can compose, according to the constraints in the statement; the maximum number of candies having $f_i = 1$ in a gift you can compose that contains the maximum possible number of candies. -----Example----- Input 3 8 1 0 4 1 2 0 4 1 5 1 6 1 3 0 2 0 4 1 1 1 1 2 1 2 1 9 2 0 2 0 4 1 4 1 4 1 7 0 7 1 7 0 7 1 Output 3 3 3 3 9 5 -----Note----- In the first query, you can include two candies of type $4$ and one candy of type $5$. All of them have $f_i = 1$ and you don't mind giving them away as part of the gift. 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: Information Search problem The posting list is a list in which there is a correspondence between the search term and the appearing document ID. For example * Hokkaido: 1, 2, 4, 9 * Sightseeing: 1, 3, 4, 7 And so on. From the above posting list, if you search for and, the document with ID 1, 4 will be hit, and if you search for or, ID 1, 2, 3, 4, 7, 9 will be hit. Here, and search means "listing elements contained in either list", or search means "listing elements contained in at least one of the lists". Since the posting list is given, output the results of and search and or search respectively. Input format n m a_1 a_2 $ \ ldots $ a_n b_1 b_2 $ \ ldots $ b_m All inputs consist of integers. The first line gives the lengths n and m of the two posting lists to search, separated by spaces. The second and third lines are given the IDs contained in their respective posting lists, separated by blanks. Constraint * 1 \ leq n, m \ leq 2 \ times 10 ^ 5 * a_i <a_j (i <j) * b_i <b_j (i <j) * 1 \ leq a_i, b_i \ leq 10 ^ 9 Output format Let A be the number of hits in the and search, and B be the number of hits in the or search. Print on the first line in the order A B, separated by blanks. Output the IDs hit by and search on the following line A in ascending order. Output the IDs hit by or search on the following B line in ascending order. Input example 1 4 4 1 2 4 9 1 3 4 7 Output example 1 2 6 1 Four 1 2 3 Four 7 9 Input example 2 4 4 1 3 5 7 2 4 6 8 Output example 2 0 8 1 2 3 Four Five 6 7 8 Input example 3 3 5 one two Three 1 2 3 4 5 Output example 3 3 5 1 2 3 1 2 3 Four Five Example Input 4 4 1 2 4 9 1 3 4 7 Output 2 6 1 4 1 2 3 4 7 9 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. Tavas is a strange creature. Usually "zzz" comes out of people's mouth while sleeping, but string s of length n comes out from Tavas' mouth instead. <image> Today Tavas fell asleep in Malekas' place. While he was sleeping, Malekas did a little process on s. Malekas has a favorite string p. He determined all positions x1 < x2 < ... < xk where p matches s. More formally, for each xi (1 ≤ i ≤ k) he condition sxisxi + 1... sxi + \|p\| - 1 = p is fullfilled. Then Malekas wrote down one of subsequences of x1, x2, ... xk (possibly, he didn't write anything) on a piece of paper. Here a sequence b is a subsequence of sequence a if and only if we can turn a into b by removing some of its elements (maybe no one of them or all). After Tavas woke up, Malekas told him everything. He couldn't remember string s, but he knew that both p and s only contains lowercase English letters and also he had the subsequence he had written on that piece of paper. Tavas wonders, what is the number of possible values of s? He asked SaDDas, but he wasn't smart enough to solve this. So, Tavas asked you to calculate this number for him. Answer can be very large, so Tavas wants you to print the answer modulo 109 + 7. Input The first line contains two integers n and m, the length of s and the length of the subsequence Malekas wrote down (1 ≤ n ≤ 106 and 0 ≤ m ≤ n - \|p\| + 1). The second line contains string p (1 ≤ \|p\| ≤ n). The next line contains m space separated integers y1, y2, ..., ym, Malekas' subsequence (1 ≤ y1 < y2 < ... < ym ≤ n - \|p\| + 1). Output In a single line print the answer modulo 1000 000 007. Examples Input 6 2 ioi 1 3 Output 26 Input 5 2 ioi 1 2 Output 0 Note In the first sample test all strings of form "ioioi?" where the question mark replaces arbitrary English letter satisfy. Here \|x\| denotes the length of string x. Please note that it's possible that there is no such string (answer is 0). 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 $n$ monsters standing in a row numbered from $1$ to $n$. The $i$-th monster has $h_i$ health points (hp). You have your attack power equal to $a$ hp and your opponent has his attack power equal to $b$ hp. You and your opponent are fighting these monsters. Firstly, you and your opponent go to the first monster and fight it till his death, then you and your opponent go the second monster and fight it till his death, and so on. A monster is considered dead if its hp is less than or equal to $0$. The fight with a monster happens in turns. You hit the monster by $a$ hp. If it is dead after your hit, you gain one point and you both proceed to the next monster. Your opponent hits the monster by $b$ hp. If it is dead after his hit, nobody gains a point and you both proceed to the next monster. You have some secret technique to force your opponent to skip his turn. You can use this technique at most $k$ times in total (for example, if there are two monsters and $k=4$, then you can use the technique $2$ times on the first monster and $1$ time on the second monster, but not $2$ times on the first monster and $3$ times on the second monster). Your task is to determine the maximum number of points you can gain if you use the secret technique optimally. -----Input----- The first line of the input contains four integers $n, a, b$ and $k$ ($1 \le n \le 2 \cdot 10^5, 1 \le a, b, k \le 10^9$) — the number of monsters, your attack power, the opponent's attack power and the number of times you can use the secret technique. The second line of the input contains $n$ integers $h_1, h_2, \dots, h_n$ ($1 \le h_i \le 10^9$), where $h_i$ is the health points of the $i$-th monster. -----Output----- Print one integer — the maximum number of points you can gain if you use the secret technique optimally. -----Examples----- Input 6 2 3 3 7 10 50 12 1 8 Output 5 Input 1 1 100 99 100 Output 1 Input 7 4 2 1 1 3 5 4 2 7 6 Output 6 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. Digits Are Not Just Characters Mr. Manuel Majorana Minore made a number of files with numbers in their names. He wants to have a list of the files, but the file listing command commonly used lists them in an order different from what he prefers, interpreting digit sequences in them as ASCII code sequences, not as numbers. For example, the files file10, file20 and file3 are listed in this order. Write a program which decides the orders of file names interpreting digit sequences as numeric values. Each file name consists of uppercase letters (from 'A' to 'Z'), lowercase letters (from 'a' to 'z'), and digits (from '0' to '9'). A file name is looked upon as a sequence of items, each being either a letter or a number. Each single uppercase or lowercase letter forms a letter item. Each consecutive sequence of digits forms a number item. Two item are ordered as follows. * Number items come before letter items. * Two letter items are ordered by their ASCII codes. * Two number items are ordered by their values when interpreted as decimal numbers. Two file names are compared item by item, starting from the top, and the order of the first different corresponding items decides the order of the file names. If one of them, say $A$, has more items than the other, $B$, and all the items of $B$ are the same as the corresponding items of $A$, $B$ should come before. For example, three file names in Sample Input 1, file10, file20, and file3 all start with the same sequence of four letter items f, i, l, and e, followed by a number item, 10, 20, and 3, respectively. Comparing numeric values of these number items, they are ordered as file3 $<$ file10 $<$ file20. Input The input consists of a single test case of the following format. $n$ $s_0$ $s_1$ : $s_n$ The integer $n$ in the first line gives the number of file names ($s_1$ through $s_n$) to be compared with the file name given in the next line ($s_0$). Here, $n$ satisfies $1 \leq n \leq 1000$. The following $n + 1$ lines are file names, $s_0$ through $s_n$, one in each line. They have at least one and no more than nine characters. Each of the characters is either an uppercase letter, a lowercase letter, or a digit. Sequences of digits in the file names never start with a digit zero (0). Output For each of the file names, $s_1$ through $s_n$, output one line with a character indicating whether it should come before $s_0$ or not. The character should be "-" if it is to be listed before $s_0$; otherwise, it should be "+", including cases where two names are identical. Sample Input 1 2 file10 file20 file3 Sample Output 1 + - Sample Input 2 11 X52Y X X5 X52 X52Y X52Y6 32 ABC XYZ x51y X8Y X222 Sample Output 2 - - - + + - - + + - + Example Input 2 file10 file20 file3 Output + - Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. This is a harder version of the problem with bigger constraints. Korney Korneevich dag up an array a of length n. Korney Korneevich has recently read about the operation [bitwise XOR](https://en.wikipedia.org/wiki/Bitwise_operation#XOR), so he wished to experiment with it. For this purpose, he decided to find all integers x ≥ 0 such that there exists an increasing subsequence of the array a, in which the bitwise XOR of numbers is equal to x. It didn't take a long time for Korney Korneevich to find all such x, and he wants to check his result. That's why he asked you to solve this problem! A sequence s is a subsequence of a sequence b if s can be obtained from b by deletion of several (possibly, zero or all) elements. A sequence s_1, s_2, … , s_m is called increasing if s_1 < s_2 < … < s_m. Input The first line contains a single integer n (1 ≤ n ≤ 10^6). The second line contains n integers a_1, a_2, …, a_n (0 ≤ a_i ≤ 5000) — the elements of the array a. Output In the first line print a single integer k — the number of found x values. In the second line print k integers in increasing order x_1, x_2, … x_k (0 ≤ x_1 < … < x_k) — found x values. Examples Input 4 4 2 2 4 Output 4 0 2 4 6 Input 8 1 0 1 7 12 5 3 2 Output 12 0 1 2 3 4 5 6 7 10 11 12 13 Note In the first test case: * To get value x = 0 it is possible to choose and empty subsequence * To get value x = 2 it is possible to choose a subsequence [2] * To get value x = 4 it is possible to choose a subsequence [4] * To get value x = 6 it is possible to choose a subsequence [2, 4] 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 $a$ consisting of $n$ ($n \ge 3$) positive integers. It is known that in this array, all the numbers except one are the same (for example, in the array $[4, 11, 4, 4]$ all numbers except one are equal to $4$). Print the index of the element that does not equal others. The numbers in the array are numbered from one. -----Input----- The first line contains a single integer $t$ ($1 \le t \le 100$). Then $t$ test cases follow. The first line of each test case contains a single integer $n$ ($3 \le n \le 100$) — the length of the array $a$. The second line of each test case contains $n$ integers $a_1, a_2, \ldots, a_n$ ($1 \le a_i \le 100$). It is guaranteed that all the numbers except one in the $a$ array are the same. -----Output----- For each test case, output a single integer — the index of the element that is not equal to others. -----Examples----- Input 4 4 11 13 11 11 5 1 4 4 4 4 10 3 3 3 3 10 3 3 3 3 3 3 20 20 10 Output 2 1 5 3 -----Note----- None 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. Fennec is fighting with N monsters. The health of the i-th monster is H_i. Fennec can do the following two actions: - Attack: Fennec chooses one monster. That monster's health will decrease by 1. - Special Move: Fennec chooses one monster. That monster's health will become 0. There is no way other than Attack and Special Move to decrease the monsters' health. Fennec wins when all the monsters' healths become 0 or below. Find the minimum number of times Fennec needs to do Attack (not counting Special Move) before winning when she can use Special Move at most K times. -----Constraints----- - 1 \leq N \leq 2 \times 10^5 - 0 \leq K \leq 2 \times 10^5 - 1 \leq H_i \leq 10^9 - All values in input are integers. -----Input----- Input is given from Standard Input in the following format: N K H_1 ... H_N -----Output----- Print the minimum number of times Fennec needs to do Attack (not counting Special Move) before winning. -----Sample Input----- 3 1 4 1 5 -----Sample Output----- 5 By using Special Move on the third monster, and doing Attack four times on the first monster and once on the second monster, Fennec can win with five Attacks. Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. One day, Twilight Sparkle is interested in how to sort a sequence of integers a_1, a_2, ..., a_{n} in non-decreasing order. Being a young unicorn, the only operation she can perform is a unit shift. That is, she can move the last element of the sequence to its beginning:a_1, a_2, ..., a_{n} → a_{n}, a_1, a_2, ..., a_{n} - 1. Help Twilight Sparkle to calculate: what is the minimum number of operations that she needs to sort the sequence? -----Input----- The first line contains an integer n (2 ≤ n ≤ 10^5). The second line contains n integer numbers a_1, a_2, ..., a_{n} (1 ≤ a_{i} ≤ 10^5). -----Output----- If it's impossible to sort the sequence output -1. Otherwise output the minimum number of operations Twilight Sparkle needs to sort it. -----Examples----- Input 2 2 1 Output 1 Input 3 1 3 2 Output -1 Input 2 1 2 Output 0 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 sitting on an extremely boring math class. To have fun, he took a piece of paper and wrote out n numbers on a single line. After that, Vasya began to write out different ways to put pluses ("+") in the line between certain digits in the line so that the result was a correct arithmetic expression; formally, no two pluses in such a partition can stand together (between any two adjacent pluses there must be at least one digit), and no plus can stand at the beginning or the end of a line. For example, in the string 100500, ways 100500 (add no pluses), 1+00+500 or 10050+0 are correct, and ways 100++500, +1+0+0+5+0+0 or 100500+ are incorrect. The lesson was long, and Vasya has written all the correct ways to place exactly k pluses in a string of digits. At this point, he got caught having fun by a teacher and he was given the task to calculate the sum of all the resulting arithmetic expressions by the end of the lesson (when calculating the value of an expression the leading zeros should be ignored). As the answer can be large, Vasya is allowed to get only its remainder modulo 109 + 7. Help him! Input The first line contains two integers, n and k (0 ≤ k < n ≤ 105). The second line contains a string consisting of n digits. Output Print the answer to the problem modulo 109 + 7. Examples Input 3 1 108 Output 27 Input 3 2 108 Output 9 Note In the first sample the result equals (1 + 08) + (10 + 8) = 27. In the second sample the result equals 1 + 0 + 8 = 9. 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. Read problems statements in Mandarin Chinese and Russian. Chef has N simple polygons (non self intersecting polygons) in which no two of them intersect with each other. For any two polygons P_{1}, P_{2}, either P_{1} lies inside P_{2} or vice versa. Chef wants you to count number of polygons lying strictly inside each of the polygons. ------ Input ------ First line of the input contains an integer T denoting the number of test cases. First line of each test case contains a single integer N denoting the number of polygons. The description of N polygons is as follows: The first line contains an integer M_{i} denoting the number of vertices in the i^{th} polygon The second line contains M_{i} pairs of integers X_{i, j}, Y_{i, j} representing coordinates of vertices of i^{th} polygon in clockwise or counterclockwise order ------ Output ------ For each test case, output a single line containing N space-separated integers such that i^{th} of them represents number of polygons lying inside the i^{th} polygon. ------ Constraints ------ $1 ≤ T ≤ 10^{5}$ $2 ≤ N ≤ 10^{5}$ $3 ≤ M_{i} ≤ 10^{5}$ $The sum of M_{i} (or total amount of given points) over all test cases in one test file does not exceed 2*10^{5}$ $Absolute value of each coordinate doesn't exceed 10^{9}$ Subtask 1: (10 points) $T ≤ 6$ $2 ≤ N ≤ 10$ $3 ≤ M_{i} ≤ 10$ $Absolute value of each coordinate doesn't exceed 30$ Subtask 2: (20 points) $All polygons are convex polygons .$ Subtask 3: (70 points) $Original constraints$ ----- Sample Input 1 ------ 1 3 6 -2 2 -1 1 2 2 2 -1 1 -2 -2 -2 3 -1 -1 1 -1 1 1 4 3 3 -3 3 -3 -3 3 -3 ----- Sample Output 1 ------ 1 0 2 ----- explanation 1 ------ In the picture the first polygon is marked in green, second - in red and third in blue color. 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 Free Meteor Association (FMA) has got a problem: as meteors are moving, the Universal Cosmic Descriptive Humorous Program (UCDHP) needs to add a special module that would analyze this movement. UCDHP stores some secret information about meteors as an n × m table with integers in its cells. The order of meteors in the Universe is changing. That's why the main UCDHP module receives the following queries: * The query to swap two table rows; * The query to swap two table columns; * The query to obtain a secret number in a particular table cell. As the main UCDHP module is critical, writing the functional of working with the table has been commissioned to you. Input The first line contains three space-separated integers n, m and k (1 ≤ n, m ≤ 1000, 1 ≤ k ≤ 500000) — the number of table columns and rows and the number of queries, correspondingly. Next n lines contain m space-separated numbers each — the initial state of the table. Each number p in the table is an integer and satisfies the inequality 0 ≤ p ≤ 106. Next k lines contain queries in the format "si xi yi", where si is one of the characters "с", "r" or "g", and xi, yi are two integers. * If si = "c", then the current query is the query to swap columns with indexes xi and yi (1 ≤ x, y ≤ m, x ≠ y); * If si = "r", then the current query is the query to swap rows with indexes xi and yi (1 ≤ x, y ≤ n, x ≠ y); * If si = "g", then the current query is the query to obtain the number that located in the xi-th row and in the yi-th column (1 ≤ x ≤ n, 1 ≤ y ≤ m). The table rows are considered to be indexed from top to bottom from 1 to n, and the table columns — from left to right from 1 to m. Output For each query to obtain a number (si = "g") print the required number. Print the answers to the queries in the order of the queries in the input. Examples Input 3 3 5 1 2 3 4 5 6 7 8 9 g 3 2 r 3 2 c 2 3 g 2 2 g 3 2 Output 8 9 6 Input 2 3 3 1 2 4 3 1 5 c 2 1 r 1 2 g 1 3 Output 5 Note Let's see how the table changes in the second test case. After the first operation is fulfilled, the table looks like that: 2 1 4 1 3 5 After the second operation is fulfilled, the table looks like that: 1 3 5 2 1 4 So the answer to the third query (the number located in the first row and in the third column) will be 5. 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 a lover of bacteria. You want to raise some bacteria in a box. Initially, the box is empty. Each morning, you can put any number of bacteria into the box. And each night, every bacterium in the box will split into two bacteria. You hope to see exactly x bacteria in the box at some moment. What is the minimum number of bacteria you need to put into the box across those days? -----Input----- The only line containing one integer x (1 ≤ x ≤ 10^9). -----Output----- The only line containing one integer: the answer. -----Examples----- Input 5 Output 2 Input 8 Output 1 -----Note----- For the first sample, we can add one bacterium in the box in the first day morning and at the third morning there will be 4 bacteria in the box. Now we put one more resulting 5 in the box. We added 2 bacteria in the process so the answer is 2. For the second sample, we can put one in the first morning and in the 4-th morning there will be 8 in the box. So the answer is 1. Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. <image> There is a container that is bifurcated as shown in the figure. Drop 10 balls numbered from 1 to 10 through the opening A of the container and place the balls in the left cylinder B or the right cylinder C. Since plate D can rotate left and right around the fulcrum E, you can move plate D to decide whether to put it in cylinder B or cylinder C. Gives a sequence of balls to drop from opening A. Put them in cylinder B or cylinder C in order. At this time, create a program that outputs YES if both cylinder B and cylinder C can arrange large balls on the small numbered balls, and NO if they cannot be arranged. However, the order of the balls cannot be changed in the container. In addition, it is assumed that they can be put in the same cylinder in succession, and that there is enough room for all 10 balls in both cylinders B and C. Input Given multiple datasets. The number of datasets N is given in the first line. Then you are given a dataset of N rows. Each dataset is given 10 numbers, starting from the left, separated by blanks. Output Print YES or NO on one line for each dataset. Example Input 2 3 1 4 2 5 6 7 8 9 10 10 9 8 7 6 5 4 3 2 1 Output YES NO Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Let s be some string consisting of symbols "0" or "1". Let's call a string t a substring of string s, if there exists such number 1 ≤ l ≤ \|s\| - \|t\| + 1 that t = s_l s_{l+1} … s_{l + \|t\| - 1}. Let's call a substring t of string s unique, if there exist only one such l. For example, let s = "1010111". A string t = "010" is an unique substring of s, because l = 2 is the only one suitable number. But, for example t = "10" isn't a unique substring of s, because l = 1 and l = 3 are suitable. And for example t ="00" at all isn't a substring of s, because there is no suitable l. Today Vasya solved the following problem at the informatics lesson: given a string consisting of symbols "0" and "1", the task is to find the length of its minimal unique substring. He has written a solution to this problem and wants to test it. He is asking you to help him. You are given 2 positive integers n and k, such that (n mod 2) = (k mod 2), where (x mod 2) is operation of taking remainder of x by dividing on 2. Find any string s consisting of n symbols "0" or "1", such that the length of its minimal unique substring is equal to k. Input The first line contains two integers n and k, separated by spaces (1 ≤ k ≤ n ≤ 100 000, (k mod 2) = (n mod 2)). Output Print a string s of length n, consisting of symbols "0" and "1". Minimal length of the unique substring of s should be equal to k. You can find any suitable string. It is guaranteed, that there exists at least one such string. Examples Input 4 4 Output 1111 Input 5 3 Output 01010 Input 7 3 Output 1011011 Note In the first test, it's easy to see, that the only unique substring of string s = "1111" is all string s, which has length 4. In the second test a string s = "01010" has minimal unique substring t ="101", which has length 3. In the third test a string s = "1011011" has minimal unique substring t ="110", which has length 3. 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. "QAQ" is a word to denote an expression of crying. Imagine "Q" as eyes with tears and "A" as a mouth. Now Diamond has given Bort a string consisting of only uppercase English letters of length n. There is a great number of "QAQ" in the string (Diamond is so cute!). $8$ illustration by 猫屋 https://twitter.com/nekoyaliu Bort wants to know how many subsequences "QAQ" are in the string Diamond has given. Note that the letters "QAQ" don't have to be consecutive, but the order of letters should be exact. -----Input----- The only line contains a string of length n (1 ≤ n ≤ 100). It's guaranteed that the string only contains uppercase English letters. -----Output----- Print a single integer — the number of subsequences "QAQ" in the string. -----Examples----- Input QAQAQYSYIOIWIN Output 4 Input QAQQQZZYNOIWIN Output 3 -----Note----- In the first example there are 4 subsequences "QAQ": "QAQAQYSYIOIWIN", "QAQAQYSYIOIWIN", "QAQAQYSYIOIWIN", "QAQAQYSYIOIWIN". 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. PolandBall is a young, clever Ball. He is interested in prime numbers. He has stated a following hypothesis: "There exists such a positive integer n that for each positive integer m number n·m + 1 is a prime number". Unfortunately, PolandBall is not experienced yet and doesn't know that his hypothesis is incorrect. Could you prove it wrong? Write a program that finds a counterexample for any n. -----Input----- The only number in the input is n (1 ≤ n ≤ 1000) — number from the PolandBall's hypothesis. -----Output----- Output such m that n·m + 1 is not a prime number. Your answer will be considered correct if you output any suitable m such that 1 ≤ m ≤ 10^3. It is guaranteed the the answer exists. -----Examples----- Input 3 Output 1 Input 4 Output 2 -----Note----- A prime number (or a prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself. For the first sample testcase, 3·1 + 1 = 4. We can output 1. In the second sample testcase, 4·1 + 1 = 5. We cannot output 1 because 5 is prime. However, m = 2 is okay since 4·2 + 1 = 9, which is not a prime number. 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 nth triangular number is defined as the sum of the first n positive integers. The nth tetrahedral number is defined as the sum of the first n triangular numbers. It is easy to show that the nth tetrahedral number is equal to n(n+1)(n+2) ⁄ 6. For example, the 5th tetrahedral number is 1+(1+2)+(1+2+3)+(1+2+3+4)+(1+2+3+4+5) = 5×6×7 ⁄ 6 = 35. The first 5 triangular numbers 1, 3, 6, 10, 15 Tr\[1\] Tr\[2\] Tr\[3\] Tr\[4\] Tr\[5\] The first 5 tetrahedral numbers 1, 4, 10, 20, 35 Tet\[1\] Tet\[2\] Tet\[3\] Tet\[4\] Tet\[5\] In 1850, Sir Frederick Pollock, 1st Baronet, who was not a professional mathematician but a British lawyer and Tory (currently known as Conservative) politician, conjectured that every positive integer can be represented as the sum of at most five tetrahedral numbers. Here, a tetrahedral number may occur in the sum more than once and, in such a case, each occurrence is counted separately. The conjecture has been open for more than one and a half century. Your mission is to write a program to verify Pollock's conjecture for individual integers. Your program should make a calculation of the least number of tetrahedral numbers to represent each input integer as their sum. In addition, for some unknown reason, your program should make a similar calculation with only odd tetrahedral numbers available. For example, one can represent 40 as the sum of 2 tetrahedral numbers, 4×5×6 ⁄ 6 + 4×5×6 ⁄ 6, but 40 itself is not a tetrahedral number. One can represent 40 as the sum of 6 odd tetrahedral numbers, 5×6×7 ⁄ 6 + 1×2×3 ⁄ 6 + 1×2×3 ⁄ 6 + 1×2×3 ⁄ 6 + 1×2×3 ⁄ 6 + 1×2×3 ⁄ 6, but cannot represent as the sum of fewer odd tetrahedral numbers. Thus, your program should report 2 and 6 if 40 is given. Input The input is a sequence of lines each of which contains a single positive integer less than 106. The end of the input is indicated by a line containing a single zero. Output For each input positive integer, output a line containing two integers separated by a space. The first integer should be the least number of tetrahedral numbers to represent the input integer as their sum. The second integer should be the least number of odd tetrahedral numbers to represent the input integer as their sum. No extra characters should appear in the output. Sample Input 40 14 5 165 120 103 106 139 0 Output for the Sample Input 2 6 2 14 2 5 1 1 1 18 5 35 4 4 3 37 Example Input 40 14 5 165 120 103 106 139 0 Output 2 6 2 14 2 5 1 1 1 18 5 35 4 4 3 37 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. Vova is again playing some computer game, now an RPG. In the game Vova's character received a quest: to slay the fearsome monster called Modcrab. After two hours of playing the game Vova has tracked the monster and analyzed its tactics. The Modcrab has h_2 health points and an attack power of a_2. Knowing that, Vova has decided to buy a lot of strong healing potions and to prepare for battle. Vova's character has h_1 health points and an attack power of a_1. Also he has a large supply of healing potions, each of which increases his current amount of health points by c_1 when Vova drinks a potion. All potions are identical to each other. It is guaranteed that c_1 > a_2. The battle consists of multiple phases. In the beginning of each phase, Vova can either attack the monster (thus reducing its health by a_1) or drink a healing potion (it increases Vova's health by c_1; Vova's health can exceed h_1). Then, if the battle is not over yet, the Modcrab attacks Vova, reducing his health by a_2. The battle ends when Vova's (or Modcrab's) health drops to 0 or lower. It is possible that the battle ends in a middle of a phase after Vova's attack. Of course, Vova wants to win the fight. But also he wants to do it as fast as possible. So he wants to make up a strategy that will allow him to win the fight after the minimum possible number of phases. Help Vova to make up a strategy! You may assume that Vova never runs out of healing potions, and that he can always win. -----Input----- The first line contains three integers h_1, a_1, c_1 (1 ≤ h_1, a_1 ≤ 100, 2 ≤ c_1 ≤ 100) — Vova's health, Vova's attack power and the healing power of a potion. The second line contains two integers h_2, a_2 (1 ≤ h_2 ≤ 100, 1 ≤ a_2 < c_1) — the Modcrab's health and his attack power. -----Output----- In the first line print one integer n denoting the minimum number of phases required to win the battle. Then print n lines. i-th line must be equal to HEAL if Vova drinks a potion in i-th phase, or STRIKE if he attacks the Modcrab. The strategy must be valid: Vova's character must not be defeated before slaying the Modcrab, and the monster's health must be 0 or lower after Vova's last action. If there are multiple optimal solutions, print any of them. -----Examples----- Input 10 6 100 17 5 Output 4 STRIKE HEAL STRIKE STRIKE Input 11 6 100 12 5 Output 2 STRIKE STRIKE -----Note----- In the first example Vova's character must heal before or after his first attack. Otherwise his health will drop to zero in 2 phases while he needs 3 strikes to win. In the second example no healing needed, two strikes are enough to get monster to zero health and win with 6 health left. Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. $n$ fishermen have just returned from a fishing vacation. The $i$-th fisherman has caught a fish of weight $a_i$. Fishermen are going to show off the fish they caught to each other. To do so, they firstly choose an order in which they show their fish (each fisherman shows his fish exactly once, so, formally, the order of showing fish is a permutation of integers from $1$ to $n$). Then they show the fish they caught according to the chosen order. When a fisherman shows his fish, he might either become happy, become sad, or stay content. Suppose a fisherman shows a fish of weight $x$, and the maximum weight of a previously shown fish is $y$ ($y = 0$ if that fisherman is the first to show his fish). Then: if $x \ge 2y$, the fisherman becomes happy; if $2x \le y$, the fisherman becomes sad; if none of these two conditions is met, the fisherman stays content. Let's call an order in which the fishermen show their fish emotional if, after all fishermen show their fish according to this order, each fisherman becomes either happy or sad. Calculate the number of emotional orders modulo $998244353$. -----Input----- The first line contains one integer $n$ ($2 \le n \le 5000$). The second line contains $n$ integers $a_1$, $a_2$, ..., $a_n$ ($1 \le a_i \le 10^9$). -----Output----- Print one integer — the number of emotional orders, taken modulo $998244353$. -----Examples----- Input 4 1 1 4 9 Output 20 Input 4 4 3 2 1 Output 0 Input 3 4 2 1 Output 6 Input 8 42 1337 13 37 420 666 616 97 Output 19200 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 program will receive an array of complex numbers represented as strings. Your task is to write the `complexSum` function which have to return the sum as a string. Complex numbers can be written in the form of `a+bi`, such as `2-3i` where `2` is the real part, `3` is the imaginary part, and `i` is the "imaginary unit". When you add two complex numbers, the real and the imaginary part needs to be added separately,so for example `2+3i + 5-i = (2+5)+(3i-i) = 7+2i` Both the complex and the imaginary part can be 0, so `123`, `-2i` or `i` are also complex numbers. Complex numbers must be returned in their shortest form, so e.g. `0+1*i` should be just `i`, and `10+0i` should be `10`. This is also how you will get them! For simplicity, the coefficients will always be integers. If the array is empty, return `0`. Have fun! :) 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 playing a computer game, where you lead a party of $m$ soldiers. Each soldier is characterised by his agility $a_i$. The level you are trying to get through can be represented as a straight line segment from point $0$ (where you and your squad is initially located) to point $n + 1$ (where the boss is located). The level is filled with $k$ traps. Each trap is represented by three numbers $l_i$, $r_i$ and $d_i$. $l_i$ is the location of the trap, and $d_i$ is the danger level of the trap: whenever a soldier with agility lower than $d_i$ steps on a trap (that is, moves to the point $l_i$), he gets instantly killed. Fortunately, you can disarm traps: if you move to the point $r_i$, you disarm this trap, and it no longer poses any danger to your soldiers. Traps don't affect you, only your soldiers. You have $t$ seconds to complete the level — that is, to bring some soldiers from your squad to the boss. Before the level starts, you choose which soldiers will be coming with you, and which soldiers won't be. After that, you have to bring all of the chosen soldiers to the boss. To do so, you may perform the following actions: if your location is $x$, you may move to $x + 1$ or $x - 1$. This action consumes one second; if your location is $x$ and the location of your squad is $x$, you may move to $x + 1$ or to $x - 1$ with your squad in one second. You may not perform this action if it puts some soldier in danger (i. e. the point your squad is moving into contains a non-disarmed trap with $d_i$ greater than agility of some soldier from the squad). This action consumes one second; if your location is $x$ and there is a trap $i$ with $r_i = x$, you may disarm this trap. This action is done instantly (it consumes no time). Note that after each action both your coordinate and the coordinate of your squad should be integers. You have to choose the maximum number of soldiers such that they all can be brought from the point $0$ to the point $n + 1$ (where the boss waits) in no more than $t$ seconds. -----Input----- The first line contains four integers $m$, $n$, $k$ and $t$ ($1 \le m, n, k, t \le 2 \cdot 10^5$, $n < t$) — the number of soldiers, the number of integer points between the squad and the boss, the number of traps and the maximum number of seconds you may spend to bring the squad to the boss, respectively. The second line contains $m$ integers $a_1$, $a_2$, ..., $a_m$ ($1 \le a_i \le 2 \cdot 10^5$), where $a_i$ is the agility of the $i$-th soldier. Then $k$ lines follow, containing the descriptions of traps. Each line contains three numbers $l_i$, $r_i$ and $d_i$ ($1 \le l_i \le r_i \le n$, $1 \le d_i \le 2 \cdot 10^5$) — the location of the trap, the location where the trap can be disarmed, and its danger level, respectively. -----Output----- Print one integer — the maximum number of soldiers you may choose so that you may bring them all to the boss in no more than $t$ seconds. -----Example----- Input 5 6 4 14 1 2 3 4 5 1 5 2 1 2 5 2 3 5 3 5 3 Output 3 -----Note----- In the first example you may take soldiers with agility $3$, $4$ and $5$ with you. The course of action is as follows: go to $2$ without your squad; disarm the trap $2$; go to $3$ without your squad; disartm the trap $3$; go to $0$ without your squad; go to $7$ with your squad. The whole plan can be executed in $13$ seconds. 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. # Task A masked number is a string that consists of digits and one asterisk (``) that should be replaced by exactly one digit. Given a masked number `s`, find all the possible options to replace the asterisk with a digit to produce an integer divisible by 6. # Input/Output `[input]` string `s` A masked number. `1 ≤ inputString.length ≤ 10000.` `[output]` a string array Sorted array of strings representing all non-negative integers that correspond to the given mask and are divisible by 6. # Example For `s = "10"`, the output should be `["120", "150", "180"].` For `s = "1"`, the output should be `[].` For `s = "12345678901234567890123456780"`, the output should be ``` [ "123456789012345678901234567800", "123456789012345678901234567830", "123456789012345678901234567860", "123456789012345678901234567890"]``` As you can see, the masked number may be very large ;-) 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 $n$ dormitories in Berland State University, they are numbered with integers from $1$ to $n$. Each dormitory consists of rooms, there are $a_i$ rooms in $i$-th dormitory. The rooms in $i$-th dormitory are numbered from $1$ to $a_i$. A postman delivers letters. Sometimes there is no specific dormitory and room number in it on an envelope. Instead of it only a room number among all rooms of all $n$ dormitories is written on an envelope. In this case, assume that all the rooms are numbered from $1$ to $a_1 + a_2 + \dots + a_n$ and the rooms of the first dormitory go first, the rooms of the second dormitory go after them and so on. For example, in case $n=2$, $a_1=3$ and $a_2=5$ an envelope can have any integer from $1$ to $8$ written on it. If the number $7$ is written on an envelope, it means that the letter should be delivered to the room number $4$ of the second dormitory. For each of $m$ letters by the room number among all $n$ dormitories, determine the particular dormitory and the room number in a dormitory where this letter should be delivered. -----Input----- The first line contains two integers $n$ and $m$ $(1 \le n, m \le 2 \cdot 10^{5})$ — the number of dormitories and the number of letters. The second line contains a sequence $a_1, a_2, \dots, a_n$ $(1 \le a_i \le 10^{10})$, where $a_i$ equals to the number of rooms in the $i$-th dormitory. The third line contains a sequence $b_1, b_2, \dots, b_m$ $(1 \le b_j \le a_1 + a_2 + \dots + a_n)$, where $b_j$ equals to the room number (among all rooms of all dormitories) for the $j$-th letter. All $b_j$ are given in increasing order. -----Output----- Print $m$ lines. For each letter print two integers $f$ and $k$ — the dormitory number $f$ $(1 \le f \le n)$ and the room number $k$ in this dormitory $(1 \le k \le a_f)$ to deliver the letter. -----Examples----- Input 3 6 10 15 12 1 9 12 23 26 37 Output 1 1 1 9 2 2 2 13 3 1 3 12 Input 2 3 5 10000000000 5 6 9999999999 Output 1 5 2 1 2 9999999994 -----Note----- In the first example letters should be delivered in the following order: the first letter in room $1$ of the first dormitory the second letter in room $9$ of the first dormitory the third letter in room $2$ of the second dormitory the fourth letter in room $13$ of the second dormitory the fifth letter in room $1$ of the third dormitory the sixth letter in room $12$ of the third dormitory 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 night wasn't easy on Vasya. His favorite team lost, and he didn't find himself victorious either — although he played perfectly, his teammates let him down every time. He had to win at least one more time, but the losestreak only grew longer and longer... It's no wonder he didn't get any sleep this night at all. In the morning, Vasya was waiting the bus to the university on the bus stop. Vasya's thoughts were hazy and so he couldn't remember the right bus' number quite right and got onto the bus with the number $n$. In the bus, Vasya thought that he could get the order of the digits in the number of the bus wrong. Futhermore, he could "see" some digits several times, but the digits he saw were definitely in the real number of the bus. For example, if Vasya saw the number 2028, it could mean that the real bus number could be 2028, 8022, 2820 or just 820. However, numbers 80, 22208, 52 definitely couldn't be the number of the bus. Also, real bus number couldn't start with the digit 0, this meaning that, for example, number 082 couldn't be the real bus number too. Given $n$, determine the total number of possible bus number variants. -----Input----- The first line contains one integer $n$ ($1 \leq n \leq 10^{18}$) — the number of the bus that was seen by Vasya. It is guaranteed that this number does not start with $0$. -----Output----- Output a single integer — the amount of possible variants of the real bus number. -----Examples----- Input 97 Output 2 Input 2028 Output 13 -----Note----- In the first sample, only variants $97$ and $79$ are possible. In the second sample, the variants (in the increasing order) are the following: $208$, $280$, $802$, $820$, $2028$, $2082$, $2208$, $2280$, $2802$, $2820$, $8022$, $8202$, $8220$. 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. Kenkoooo is planning a trip in Republic of Snuke. In this country, there are n cities and m trains running. The cities are numbered 1 through n, and the i-th train connects City u_i and v_i bidirectionally. Any city can be reached from any city by changing trains. Two currencies are used in the country: yen and snuuk. Any train fare can be paid by both yen and snuuk. The fare of the i-th train is a_i yen if paid in yen, and b_i snuuk if paid in snuuk. In a city with a money exchange office, you can change 1 yen into 1 snuuk. However, when you do a money exchange, you have to change all your yen into snuuk. That is, if Kenkoooo does a money exchange when he has X yen, he will then have X snuuk. Currently, there is a money exchange office in every city, but the office in City i will shut down in i years and can never be used in and after that year. Kenkoooo is planning to depart City s with 10^{15} yen in his pocket and head for City t, and change his yen into snuuk in some city while traveling. It is acceptable to do the exchange in City s or City t. Kenkoooo would like to have as much snuuk as possible when he reaches City t by making the optimal choices for the route to travel and the city to do the exchange. For each i=0,...,n-1, find the maximum amount of snuuk that Kenkoooo has when he reaches City t if he goes on a trip from City s to City t after i years. You can assume that the trip finishes within the year. Constraints * 2 \leq n \leq 10^5 * 1 \leq m \leq 10^5 * 1 \leq s,t \leq n * s \neq t * 1 \leq u_i < v_i \leq n * 1 \leq a_i,b_i \leq 10^9 * If i\neq j, then u_i \neq u_j or v_i \neq v_j. * Any city can be reached from any city by changing trains. * All values in input are integers. Input Input is given from Standard Input in the following format: n m s t u_1 v_1 a_1 b_1 : u_m v_m a_m b_m Output Print n lines. In the i-th line, print the maximum amount of snuuk that Kenkoooo has when he reaches City t if he goes on a trip from City s to City t after i-1 years. Examples Input 4 3 2 3 1 4 1 100 1 2 1 10 1 3 20 1 Output 999999999999998 999999999999989 999999999999979 999999999999897 Input 8 12 3 8 2 8 685087149 857180777 6 7 298270585 209942236 2 4 346080035 234079976 2 5 131857300 22507157 4 8 30723332 173476334 2 6 480845267 448565596 1 4 181424400 548830121 4 5 57429995 195056405 7 8 160277628 479932440 1 6 475692952 203530153 3 5 336869679 160714712 2 7 389775999 199123879 Output 999999574976994 999999574976994 999999574976994 999999574976994 999999574976994 999999574976994 999999574976994 999999574976994 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. Ghosts live in harmony and peace, they travel the space without any purpose other than scare whoever stands in their way. There are $n$ ghosts in the universe, they move in the $OXY$ plane, each one of them has its own velocity that does not change in time: $\overrightarrow{V} = V_{x}\overrightarrow{i} + V_{y}\overrightarrow{j}$ where $V_{x}$ is its speed on the $x$-axis and $V_{y}$ is on the $y$-axis. A ghost $i$ has experience value $EX_i$, which represent how many ghosts tried to scare him in his past. Two ghosts scare each other if they were in the same cartesian point at a moment of time. As the ghosts move with constant speed, after some moment of time there will be no further scaring (what a relief!) and the experience of ghost kind $GX = \sum_{i=1}^{n} EX_i$ will never increase. Tameem is a red giant, he took a picture of the cartesian plane at a certain moment of time $T$, and magically all the ghosts were aligned on a line of the form $y = a \cdot x + b$. You have to compute what will be the experience index of the ghost kind $GX$ in the indefinite future, this is your task for today. Note that when Tameem took the picture, $GX$ may already be greater than $0$, because many ghosts may have scared one another at any moment between $[-\infty, T]$. -----Input----- The first line contains three integers $n$, $a$ and $b$ ($1 \leq n \leq 200000$, $1 \leq \|a\| \leq 10^9$, $0 \le \|b\| \le 10^9$) — the number of ghosts in the universe and the parameters of the straight line. Each of the next $n$ lines contains three integers $x_i$, $V_{xi}$, $V_{yi}$ ($-10^9 \leq x_i \leq 10^9$, $-10^9 \leq V_{x i}, V_{y i} \leq 10^9$), where $x_i$ is the current $x$-coordinate of the $i$-th ghost (and $y_i = a \cdot x_i + b$). It is guaranteed that no two ghosts share the same initial position, in other words, it is guaranteed that for all $(i,j)$ $x_i \neq x_j$ for $i \ne j$. -----Output----- Output one line: experience index of the ghost kind $GX$ in the indefinite future. -----Examples----- Input 4 1 1 1 -1 -1 2 1 1 3 1 1 4 -1 -1 Output 8 Input 3 1 0 -1 1 0 0 0 -1 1 -1 -2 Output 6 Input 3 1 0 0 0 0 1 0 0 2 0 0 Output 0 -----Note----- There are four collisions $(1,2,T-0.5)$, $(1,3,T-1)$, $(2,4,T+1)$, $(3,4,T+0.5)$, where $(u,v,t)$ means a collision happened between ghosts $u$ and $v$ at moment $t$. At each collision, each ghost gained one experience point, this means that $GX = 4 \cdot 2 = 8$. In the second test, all points will collide when $t = T + 1$. [Image] The red arrow represents the 1-st ghost velocity, orange represents the 2-nd ghost velocity, and blue represents the 3-rd ghost velocity. 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 drawing below we have a part of the Pascal's triangle, lines are numbered from zero (top). The left diagonal in pale blue with only numbers equal to 1 is diagonal zero, then in dark green (1, 2, 3, 4, 5, 6, 7) is diagonal 1, then in pale green (1, 3, 6, 10, 15, 21) is diagonal 2 and so on. We want to calculate the sum of the binomial coefficients on a given diagonal. The sum on diagonal 0 is 8 (we'll write it S(7, 0), 7 is the number of the line where we start, 0 is the number of the diagonal). In the same way S(7, 1) is 28, S(7, 2) is 56. Can you write a program which calculate S(n, p) where n is the line where we start and p is the number of the diagonal? The function will take n and p (with: `n >= p >= 0`) as parameters and will return the sum. ## Examples: ``` diagonal(20, 3) => 5985 diagonal(20, 4) => 20349 ``` ## Hint: When following a diagonal from top to bottom have a look at the numbers on the diagonal at its right. ## Ref: http://mathworld.wolfram.com/BinomialCoefficient.html ![alternative text](http://i.imgur.com/eUGaNvIm.jpg) 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. "Balloons should be captured efficiently", the game designer says. He is designing an oldfashioned game with two dimensional graphics. In the game, balloons fall onto the ground one after another, and the player manipulates a robot vehicle on the ground to capture the balloons. The player can control the vehicle to move left or right, or simply stay. When one of the balloons reaches the ground, the vehicle and the balloon must reside at the same position, otherwise the balloon will burst and the game ends. <image> Figure B.1: Robot vehicle and falling balloons The goal of the game is to store all the balloons into the house at the left end on the game field. The vehicle can carry at most three balloons at a time, but its speed changes according to the number of the carrying balloons. When the vehicle carries k balloons (k = 0, 1, 2, 3), it takes k+1 units of time to move one unit distance. The player will get higher score when the total moving distance of the vehicle is shorter. Your mission is to help the game designer check game data consisting of a set of balloons. Given a landing position (as the distance from the house) and a landing time of each balloon, you must judge whether a player can capture all the balloons, and answer the minimum moving distance needed to capture and store all the balloons. The vehicle starts from the house. If the player cannot capture all the balloons, you must identify the first balloon that the player cannot capture. Input The input is a sequence of datasets. Each dataset is formatted as follows. n p1 t1 . . . pn tn The first line contains an integer n, which represents the number of balloons (0 < n ≤ 40). Each of the following n lines contains two integers pi and ti (1 ≤ i ≤ n) separated by a space. pi and ti represent the position and the time when the i-th balloon reaches the ground (0 < pi ≤ 100, 0 < ti ≤ 50000). You can assume ti < tj for i < j. The position of the house is 0, and the game starts from the time 0. The sizes of the vehicle, the house, and the balloons are small enough, and should be ignored. The vehicle needs 0 time for catching the balloons or storing them into the house. The vehicle can start moving immediately after these operations. The end of the input is indicated by a line containing a zero. Output For each dataset, output one word and one integer in a line separated by a space. No extra characters should occur in the output. * If the player can capture all the balloons, output "OK" and an integer that represents the minimum moving distance of the vehicle to capture and store all the balloons. * If it is impossible for the player to capture all the balloons, output "NG" and an integer k such that the k-th balloon in the dataset is the first balloon that the player cannot capture. Example Input 2 10 100 100 270 2 10 100 100 280 3 100 150 10 360 40 450 3 100 150 10 360 40 440 2 100 10 50 200 2 100 100 50 110 1 15 10 4 1 10 2 20 3 100 90 200 0 Output OK 220 OK 200 OK 260 OK 280 NG 1 NG 2 NG 1 OK 188 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 have given tree consist of n vertices. Select a vertex as root vertex that satisfies the condition below. * For all vertices v_{1} and v_{2}, if distance(root, v_{1}) = distance(root, v_{2}) then degree(v_{1}) = degree(v_{2}), where degree means the number of vertices connected to that vertex, and distance means the number of edges between two vertices. Determine and find if there is such root vertex in the tree. If there are multiple answers, find any of them. Input The first line contains a single integer n (1 ≤ n ≤ 10^{5}) — the number of vertices. Each of the next n-1 lines contains two integers v_{i} and u_{i} (1 ≤ v_{i} < u_{i} ≤ n) — it means there is an edge exist between v_{i} and u_{i}. It is guaranteed that the graph forms tree. Output If there is such root vertex exists, print any of them. Otherwise, print -1. Examples Input 7 1 2 2 3 3 4 4 5 3 6 6 7 Output 3 Input 6 1 3 2 3 3 4 4 5 4 6 Output -1 Note This is the picture for the first example. 1, 5, 7 also can be a valid answer. <image> This is the picture for the second example. You can see that it's impossible to find such root vertex. <image> 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. Alica and Bob are playing a game. Initially they have a binary string $s$ consisting of only characters 0 and 1. Alice and Bob make alternating moves: Alice makes the first move, Bob makes the second move, Alice makes the third one, and so on. During each move, the current player must choose two different adjacent characters of string $s$ and delete them. For example, if $s = 1011001$ then the following moves are possible: delete $s_1$ and $s_2$: $\textbf{10}11001 \rightarrow 11001$; delete $s_2$ and $s_3$: $1\textbf{01}1001 \rightarrow 11001$; delete $s_4$ and $s_5$: $101\textbf{10}01 \rightarrow 10101$; delete $s_6$ and $s_7$: $10110\textbf{01} \rightarrow 10110$. If a player can't make any move, they lose. Both players play optimally. You have to determine if Alice can win. -----Input----- First line contains one integer $t$ ($1 \le t \le 1000$) — the number of test cases. Only line of each test case contains one string $s$ ($1 \le \|s\| \le 100$), consisting of only characters 0 and 1. -----Output----- For each test case print answer in the single line. If Alice can win print DA (YES in Russian) in any register. Otherwise print NET (NO in Russian) in any register. -----Example----- Input 3 01 1111 0011 Output DA NET NET -----Note----- In the first test case after Alice's move string $s$ become empty and Bob can not make any move. In the second test case Alice can not make any move initially. In the third test case after Alice's move string $s$ turn into $01$. Then, after Bob's move string $s$ become empty and Alice can not make any move. 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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