E. Maximum OR Popcounttime limit per test2 secondsmemory limit per test256 megabytesinputstandard inputoutputstandard outputYou are given an array of $$$n$$$ non-negative integers.You want to answer given $$$q$$$ independent scenarios. In the $$$i$$$-th scenario, you are allowed to perform the following operation at most $$$b_i$$$ times: Choose an element of the array and increase it by $$$1$$$. Your goal is to maximize the number of bits that are equal to $$$1$$$ in the bitwise OR of all numbers in the array. Find this number for each scenario.InputEach test contains multiple test cases. The first line contains the number of test cases $$$t$$$ ($$$1 \le t \le 10^3$$$). The description of the test cases follows. The first line of each test case contains two integers $$$n$$$ and $$$q$$$ ($$$1 \leq n, q \leq 10^{5}$$$) — the size of the array and the number of scenarios.The second line contains $$$n$$$ integers $$$a_1, a_2, \ldots, a_n$$$ ($$$0 \leq a_i \leq 10^{9}$$$) — the elements of the array.The $$$i$$$-th of the next $$$q$$$ lines contains a single integer $$$b_i$$$ ($$$0 \leq b_i \leq 10^{9}$$$) — the maximum number of operations allowed in the $$$i$$$-th scenario.It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$10^{5}$$$, and the sum of $$$q$$$ over all test cases does not exceed $$$10^{5}$$$.OutputFor each test case, output $$$q$$$ lines, the $$$i$$$-th of them containing a single integer — the maximum possible number of bits equal to $$$1$$$ in the bitwise OR in the $$$i$$$-th scenario.ExampleInput31 300242 21 3032 11000000000 10000000001000000000Output0122331NoteVisualizer linkIn the first test case: In the first scenario, we don't have any operations, and therefore the answer is equal to the number of $$$1$$$-bits in the bitwise OR of the original array, $$$0$$$, which has $$$0$$$ bits set in its binary representation. In the second scenario, one way of achieving $$$1$$$ bit set in the bitwise OR is increasing $$$a_1$$$ by $$$1$$$ twice, obtaining a bitwise OR of $$$2={(10)}_2$$$. It can be shown that it is the best possible value we can obtain by performing the operation at most twice. In the third scenario, one way of achieving a result of $$$2$$$ is adding $$$1$$$ to $$$a_1$$$ three times, which can be shown to be optimal. Note that you don't have to apply the operation $$$4$$$ times. In the second test case: In the first scenario, we don't have any operations, and therefore the answer is equal to the number of $$$1$$$-bits in the bitwise OR of the original array, which is $$$2$$$. In the second scenario, one way of achieving a result of $$$3$$$ is adding $$$1$$$ to $$$a_2$$$ three times, which can be shown to be optimal.