Bitwise AND and Logarithmic Double Sum

Given two non-negative integers X and Y, calculate the value of

$$ S = \sum_{i=0}^{X}\sum_{j=[i=0]}^{Y} [i\&j=0] \left\lfloor \log_2(i+j)+1 \right\rfloor \pmod{10^9+7} $$

where:

• \& denotes the bitwise AND operation.
• [A] equals 1 if A is true, and equals 0 otherwise. In particular, the inner summation index j starts at [i=0] which is 1 when i equals 0 and 0 otherwise.
• \( \lfloor x \rfloor \) denotes the floor of x, i.e. the greatest integer less than or equal to x.

For each valid pair (i, j) (with i from 0 to X and j from [i=0] to Y), if the condition i \& j = 0 holds then add \(\left\lfloor \log_2(i+j)+1 \right\rfloor\) to the sum. Finally, output the result modulo \(10^9+7\).

inputFormat

The input consists of two non-negative integers X and Y separated by a space.

outputFormat

Output a single integer, the value of the defined sum modulo \(10^9+7\).

sample

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