Ehab the Xorcist

Given 2 integers u and v, find the shortest array such that bitwise-xor of its elements is u, and the sum of its elements is v.

Input

The only line contains 2 integers u and v (0 ≤ u,v ≤ 10^{18}).

Output

If there's no array that satisfies the condition, print "-1". Otherwise:

The first line should contain one integer, n, representing the length of the desired array. The next line should contain n positive integers, the array itself. If there are multiple possible answers, print any.

Examples

Input

2 4

Output

2 3 1

Input

1 3

Output

3 1 1 1

Input

8 5

Output

-1

Input

0 0

Output

0

Note

In the first sample, 3⊕ 1 = 2 and 3 + 1 = 4. There is no valid array of smaller length.

Notice that in the fourth sample the array is empty.

inputFormat

Input

The only line contains 2 integers u and v (0 ≤ u,v ≤ 10^{18}).

outputFormat

Output

If there's no array that satisfies the condition, print "-1". Otherwise:

The first line should contain one integer, n, representing the length of the desired array. The next line should contain n positive integers, the array itself. If there are multiple possible answers, print any.

Examples

Input

2 4

Output

2 3 1

Input

1 3

Output

3 1 1 1

Input

8 5

Output

-1

Input

0 0

Output

0

Note

In the first sample, 3⊕ 1 = 2 and 3 + 1 = 4. There is no valid array of smaller length.

Notice that in the fourth sample the array is empty.

样例

2 4

2 3 1

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