Let lowbit(x)lowbit⁡(𝑥) denote the value of the lowest binary bit of x𝑥, e.g. lowbit(12)=4lowbit⁡(12)=4, lowbit(8)=8lowbit⁡(8)=8.For an array a𝑎 of length n𝑛, if an array s𝑠 of length n𝑛 satisfies sk=(∑i=k−lowbit(k)+1kai)mod998244353𝑠𝑘=(∑𝑖=𝑘−lowbit⁡(𝑘)+1𝑘𝑎𝑖)mod998244353 for all k𝑘, then s𝑠 is called the Fenwick Tree of a𝑎. Let's denote it as s=f(a)𝑠=𝑓(𝑎).For a positive integer k𝑘 and an array a𝑎, fk(a)𝑓𝑘(𝑎) is defined as follows:fk(a)={f(a)f(fk−1(a))if k=1otherwise.𝑓𝑘(𝑎)={𝑓(𝑎)if 𝑘=1𝑓(𝑓𝑘−1(𝑎))otherwise.You are given an array b𝑏 of length n𝑛 and a positive integer k𝑘. Find an array a𝑎 that satisfies 0≤ai<9982443530≤𝑎𝑖<998244353 and fk(a)=b𝑓𝑘(𝑎)=𝑏. It can be proved that an answer always exists. If there are multiple possible answers, you may print any of them.InputEach test contains multiple test cases. The first line contains the number of test cases t𝑡 (1≤t≤1041≤𝑡≤104). The description of the test cases follows.The first line of each test case contains two positive integers n𝑛 (1≤n≤2⋅1051≤𝑛≤2⋅105) and k𝑘 (1≤k≤1091≤𝑘≤109), representing the length of the array and the number of times the function f𝑓 is performed.The second line of each test case contains an array b𝑏 of length n𝑛 (0≤bi<9982443530≤𝑏𝑖<998244353).It is guaranteed that the sum of n𝑛 over all test cases does not exceed 2⋅1052⋅105.OutputFor each test case, print a single line, containing a valid array a𝑎 of length n𝑛.ExampleinputCopy28 11 2 1 4 1 2 1 86 21 4 3 17 5 16outputCopy1 1 1 1 1 1 1 11 2 3 4 5 6

A binary tree should have at least

binary tree should have at least

Square StringThe sqing value for a binary string 𝑆S of length 𝑁N is calculated as follows:Let 𝑋=0X=0 be the initial sqing value.For each 𝑖i from 11 to 𝑁N, let 𝑗j be the largest index such that 1≤𝑗<𝑖1≤j<i and 𝑆𝑗≠𝑆𝑖S j​ =S i​ .If such an index 𝑗j exists, add (𝑖−𝑗)2(i−j) 2 to 𝑋X, otherwise do nothing.For example, consider 𝑆=000110S=000110.For 𝑖=1,2,3i=1,2,3, there is no 𝑗<𝑖j<i such that 𝑆𝑗≠𝑆𝑖S j​ =S i​ .For 𝑖=4i=4 and 𝑖=5i=5, we find 𝑗=3j=3, and add (4−3)2=1(4−3) 2 =1 and (5−3)2=4(5−3) 2 =4 to the answer, respectively.For 𝑖=6i=6, we find 𝑗=5j=5 and add (6−5)2=1(6−5) 2 =1 to the answer.The sqing value is thus 1+4+1=61+4+1=6.You are given 𝑁N. There are 2𝑁2 N binary strings of length 𝑁N.Find the sum of all their sqing values.This sum can be large, so compute it modulo 109+710 9 +7.As a reminder, a binary string is a string whose characters are all either 00 or 11.Input FormatThe first line of input will contain a single integer 𝑇T, denoting the number of test cases.The first and only line of each test case contains a single integer 𝑁N — the length of the binary strings to be considered.Output FormatFor each test case, output on a new line the answer, modulo 109+710 9 +7.Constraints1≤𝑇≤10001≤T≤10001≤𝑁≤5⋅1051≤N≤5⋅10 5 The sum of 𝑁N over all test cases won't exceed 5⋅1055⋅10 5 .Sample 1:InputOutput323838216839372959Explanation:Test case 11: The possible strings are {"00","01","10","11"}{"00","01","10","11"}.The sqing value for "00""00" is 00.The sqing value for "01""01" is 11 (for 𝑖=2i=2, we find 𝑗=1j=1).The sqing value for "10""10" is 11 (for 𝑖=2i=2, we find 𝑗=1j=1).The sqing value for "11""11" is 00.Hence, the answer to this test case is 22.Test case 22: There are 88 binary strings of length 33.The sqing value for "000""000" is 00The sqing value for "001""001" is 11The sqing value for "010""010" is 22The sqing value for "011""011" is 55The sqing value for "100""100" is 55The sqing value for "101""101" is 22The sqing value for "110""110" is 11The sqing value for "111""111" is 00Hence, the answer to this test case is 1616.

A binary tree in which all its levels except possibly the last, have the maximum number of nodes and all the nodes at the last level appear as far left as possible, is known as:

Today, Cat and Fox found an array a𝑎 consisting of n𝑛 non-negative integers.Define the loneliness of a𝑎 as the smallest positive integer k𝑘 (1≤k≤n1≤𝑘≤𝑛) such that for any two positive integers i𝑖 and j𝑗 (1≤i,j≤n−k+11≤𝑖,𝑗≤𝑛−𝑘+1), the following holds:ai|ai+1|…|ai+k−1=aj|aj+1|…|aj+k−1,𝑎𝑖|𝑎𝑖+1|…|𝑎𝑖+𝑘−1=𝑎𝑗|𝑎𝑗+1|…|𝑎𝑗+𝑘−1,where x|y𝑥|𝑦 denotes the bitwise OR of x𝑥 and y𝑦. In other words, for every k𝑘 consecutive elements, their bitwise OR should be the same. Note that the loneliness of a𝑎 is well-defined, because for k=n𝑘=𝑛 the condition is satisfied.Cat and Fox want to know how lonely the array a𝑎 is. Help them calculate the loneliness of the found array.InputEach test consists of multiple test cases. The first line contains a single integer t𝑡 (1≤t≤1041≤𝑡≤104) — the number of test cases. The description of the test cases follows.The first line of each test case contains one integer n𝑛 (1≤n≤1051≤𝑛≤105) — the length of the array a𝑎.The second line of each test case contains n𝑛 integers a1,a2,…,an𝑎1,𝑎2,…,𝑎𝑛 (0≤ai<2200≤𝑎𝑖<220) — the elements of the array.It is guaranteed that the sum of n𝑛 over all test cases doesn't exceed 105105.OutputFor each test case, print one integer  — the loneliness of the given array.ExampleinputCopy71032 2 231 0 253 0 1 4 252 0 4 0 270 0 0 0 1 2 480 1 3 2 2 1 0 3outputCopy1134473