Sequence Construction

문제

Lately, the cows on Farmer John's farm have been infatuated with watching the show Apothecowry Dairies. The show revolves around a clever bovine sleuth CowCow solving problems of various kinds. Bessie found a new problem from the show, but the solution won't be revealed until the next episode in a week! Please solve the problem for her.

You are given integers $MM$ and $KK$ $(1\le M\le {10}^9,1\le K\le 31)(1\; \backslash leq\; M\; \backslash leq\; 10\; ^\; 9,\; 1\; \backslash leq\; K\; \backslash leq\; 31)$. Please choose a positive integer $NN$ and construct a sequence $aa$ of $NN$ non-negative integers such that the following conditions are satisfied:

• $1\le N\le 1001\; \backslash le\; N\; \backslash le\; 100$
• $a_1+a_2+\cdots +a_N=Ma\_1\; +\; a\_2\; +\; \backslash dots\; +\; a\_N\; =\; M$
• $\text{popcount}(a_1)\oplus \text{ popcount}(a_2)\oplus \cdots \oplus \text{ popcount}(a_N)=K\backslash text\{popcount\}(a\_1)\; \backslash oplus\; \backslash text\{\; popcount\}(a\_2)\; \backslash oplus\; \backslash dots\; \backslash oplus\; \backslash text\{\; popcount\}(a\_N)\; =\; K$

If no such sequence exists, print $-1-1$.

$†\text{ popcount}(x)\backslash dagger\; \backslash text\{\; popcount\}(x)$ is the number of bits equal to $11$ in the binary representation of the integer $xx$. For instance, the popcount of $1111$ is $33$ and the popcount of $1616$ is $11$.

$†\oplus \backslash dagger\; \backslash oplus$ is the bitwise xor operator.

The input will consist of $TT$ ($1\le T\le 5\cdot {10}^{3}1\; \backslash le\; T\; \backslash le\; 5\; \backslash cdot\; 10^3$) independent test cases.

입력

The first line contains $TT$.

The first and only line of each test case has $MM$ and $KK$.

It is guaranteed that all test cases are unique.

출력

Output the solutions for $TT$ test cases as follows:

If no answer exists, the only line for that test case should be $-1-1$.

Otherwise, the first line for that test case should be a single integer $NN$, the length of the sequence -- ($1\le N\le 1001\; \backslash le\; N\; \backslash le\; 100$).

The second line for that test case should contain $NN$ space-separated integers that satisfy the conditions -- ($0\le a_i\le M0\; \backslash le\; a\_i\; \backslash le\; M$).

3
2 1
33 5
10 5

2
2 0
3
3 23 7
-1