You are given a sequence [a1,…,an][𝑎1,…,𝑎𝑛] where each element ai𝑎𝑖 is either 00 or 11. You can apply several (possibly zero) operations to the sequence. In each operation, you select two integers 1≤l≤r≤|a|1≤𝑙≤𝑟≤|𝑎| (where |a||𝑎| is the current length of a𝑎) and replace [al,…,ar][𝑎𝑙,…,𝑎𝑟] with a single element x𝑥, where x𝑥 is the majority of [al,…,ar][𝑎𝑙,…,𝑎𝑟].Here, the majority of a sequence consisting of 00 and 11 is defined as follows: suppose there are c0𝑐0 zeros and c1𝑐1 ones in the sequence, respectively.If c0≥c1𝑐0≥𝑐1, the majority is 00.If c0<c1𝑐0<𝑐1, the majority is 11.For example, suppose a=[1,0,0,0,1,1]𝑎=[1,0,0,0,1,1]. If we select l=1,r=2𝑙=1,𝑟=2, the resulting sequence will be [0,0,0,1,1][0,0,0,1,1]. If we select l=4,r=6𝑙=4,𝑟=6, the resulting sequence will be [1,0,0,1][1,0,0,1].Determine if you can make a=[1]𝑎=[1] with a finite number of operations.InputEach test contains multiple test cases. The first line contains the number of test cases t𝑡 (1≤t≤4⋅1041≤𝑡≤4⋅104). Description of the test cases follows.The first line of each testcase contains one integer n𝑛 (1≤n≤2⋅1051≤𝑛≤2⋅105).The second line of each testcase contains a string consisting of 00 and 11, describing the sequence a𝑎.It's guaranteed that the sum of n𝑛 over all testcases does not exceed 2⋅1052⋅105.OutputFor each testcase, if it's possible to make a=[1]𝑎=[1], print YES. Otherwise, print NO. You can output the answer in any case (upper or lower). For example, the strings yEs, yes, Yes, and YES will be recognized as positive responses.ExampleinputCopy5101120191000000019000011000outputCopyNoYesNoYesNoNoteIn the fourth testcase of the example, initially a=[1,0,0,0,0,0,0,0,1]𝑎=[1,0,0,0,0,0,0,0,1]. A valid sequence of operations is:Select l=2,r=8𝑙=2,𝑟=8 and apply the operation. a𝑎 becomes [1,0,1][1,0,1].Select l=1,r=3𝑙=1,𝑟=3 and apply the operation. a𝑎 becomes [1][1].

You are given an array 𝐴A containing 𝑁N integers.Consider the following process:Let 𝑆=0S=0 initially.For each 𝑖i from 11 to 𝑁N in order, update 𝑆S to either (𝑆+𝐴𝑖)(S+A i​ ) or (𝑆×𝐴𝑖)(S×A i​ ).That is, either add 𝐴𝑖A i​ to 𝑆S or multiply 𝑆S by 𝐴𝑖A i​ .Before performing the process, you're allowed to freely rearrange the elements of 𝐴A as you like.If you choose the rearrangement of 𝐴A and the sequence of operations optimally, what's the maximum possible value of 𝑆S that you can obtain?This maximum value can be very large, so print it modulo 109+710 9 +7.Input FormatThe first line of input will contain a single integer 𝑇T, denoting the number of test cases.Each test case consists of two lines of input.The first line of each test case contains a single integer 𝑁N — the number of elements in the array.The second line contains 𝑁N space-separated integers 𝐴1,𝐴2,…,𝐴𝑁A 1​ ,A 2​ ,…,A N​ - the elements of the array.Output FormatFor each test case, output on a new line the maximum possible value of 𝑆S, modulo 109+710 9 +7.Constraints1≤𝑇≤1031≤T≤10 3 1≤𝑁≤2⋅1051≤N≤2⋅10 5 1≤𝐴𝑖≤1091≤A i​ ≤10 9 The sum of 𝑁N over all test cases won't exceed 2⋅1052⋅10 5 .Sample 1:InputOutput244 2 5 231 2 1804Explanation:Test case 11: Choose the rearrangement 𝐴=[2,2,5,4]A=[2,2,5,4]. Then,Add 𝐴1=2A 1​ =2 to 𝑆S. Now, 𝑆=2S=2.Add 𝐴2=2A 2​ =2 to 𝑆S. Now, 𝑆=4S=4.Multiply 𝑆S by 𝐴3=5A 3​ =5. Now, 𝑆=20S=20.Multiply 𝑆S by 𝐴4=4A 4​ =4. Now, 𝑆=80S=80.This is the maximum value that can be obtained.Test case 22: Choose any rearrangement and sum up all the numbers to get 1+1+2=41+1+2=4.This is the maximum value that can be obtained.

Suppose a sequence an, is defined as follows:a1 = 9/10, a2 = 10/11, an+2 = an+1an

You are given a positive integer x𝑥. Find any array of integers a0,a1,…,an−1𝑎0,𝑎1,…,𝑎𝑛−1 for which the following holds:1≤n≤321≤𝑛≤32,ai𝑎𝑖 is 11, 00, or −1−1 for all 0≤i≤n−10≤𝑖≤𝑛−1,x=∑i=0n−1ai⋅2i𝑥=∑𝑖=0𝑛−1𝑎𝑖⋅2𝑖,There does not exist an index 0≤i≤n−20≤𝑖≤𝑛−2 such that both ai≠0𝑎𝑖≠0 and ai+1≠0𝑎𝑖+1≠0.It can be proven that under the constraints of the problem, a valid array always exists.InputEach test contains multiple test cases. The first line of input contains a single integer t𝑡 (1≤t≤1041≤𝑡≤104) — the number of test cases. The description of the test cases follows.The only line of each test case contains a single positive integer x𝑥 (1≤x<2301≤𝑥<230).OutputFor each test case, output two lines.On the first line, output an integer n𝑛 (1≤n≤321≤𝑛≤32) — the length of the array a0,a1,…,an−1𝑎0,𝑎1,…,𝑎𝑛−1.On the second line, output the array a0,a1,…,an−1𝑎0,𝑎1,…,𝑎𝑛−1.If there are multiple valid arrays, you can output any of them.ExampleinputCopy71142415271119outputCopy1150 -1 0 0 160 0 0 -1 0 15-1 0 0 0 16-1 0 -1 0 0 15-1 0 -1 0 15-1 0 1 0 1NoteIn the first test case, one valid array is [1][1], since (1)⋅20=1(1)⋅20=1.In the second test case, one possible valid array is [0,−1,0,0,1][0,−1,0,0,1], since (0)⋅20+(−1)⋅21+(0)⋅22+(0)⋅23+(1)⋅24=−2+16=14(0)⋅20+(−1)⋅21+(0)⋅22+(0)⋅23+(1)⋅24=−2+16=14.

Suppose a sequence an, is defined as follows: a1 = 9/10, a2 = 10/11, an+2 = an+1an.Show that 0 < an < 1 for all (positive integers) n.

You are given two integer arrays: array a𝑎 of length n𝑛 and array b𝑏 of length n+1𝑛+1.You can perform the following operations any number of times in any order:choose any element of the array a𝑎 and increase it by 11;choose any element of the array a𝑎 and decrease it by 11;choose any element of the array a𝑎, copy it and append the copy to the end of the array a𝑎.Your task is to calculate the minimum number of aforementioned operations (possibly zero) required to transform the array a𝑎 into the array b𝑏. It can be shown that under the constraints of the problem, it is always possible.InputThe first line contains a single integer t𝑡 (1≤t≤1041≤𝑡≤104) — the number of test cases.Each test case consists of three lines:the first line contains a single integer n𝑛 (1≤n≤2⋅1051≤𝑛≤2⋅105);the second line contains n𝑛 integers a1,a2,…,an𝑎1,𝑎2,…,𝑎𝑛 (1≤ai≤1091≤𝑎𝑖≤109);the third line contains n+1𝑛+1 integers b1,b2,…,bn+1𝑏1,𝑏2,…,𝑏𝑛+1 (1≤bi≤1091≤𝑏𝑖≤109).Additional constraint on the input: the sum of n𝑛 over all test cases doesn't exceed 2⋅1052⋅105.OutputFor each test case, print a single integer — the minimum number of operations (possibly zero) required to transform the array a𝑎 into the array b𝑏.ExampleinputCopy3121 323 33 3 344 2 1 22 1 5 2 3outputCopy318NoteIn the first example, you can transform a𝑎 into b𝑏 as follows: [2]→[2,2]→[1,2]→[1,3][2]→[2,2]→[1,2]→[1,3].