101294: [AtCoder]ABC129 E - Sum Equals Xor
Memory Limit：256 MB
Time Limit：2 S
Judge Style：Text Compare
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Solved：0
Description
Score : $500$ points
Problem Statement
You are given a positive integer $L$ in base two. How many pairs of non-negative integers $(a, b)$ satisfy the following conditions?
- $a + b \leq L$
- $a + b = a \text{ XOR } b$
Since there can be extremely many such pairs, print the count modulo $10^9 + 7$.
What is XOR?
The XOR of integers $A$ and $B$, $A \text{ XOR } B$, is defined as follows:
- When $A \text{ XOR } B$ is written in base two, the digit in the $2^k$'s place ($k \geq 0$) is $1$ if either $A$ or $B$, but not both, has $1$ in the $2^k$'s place, and $0$ otherwise.
For example, $3 \text{ XOR } 5 = 6$. (In base two: $011 \text{ XOR } 101 = 110$.)
Constraints
- $L$ is given in base two, without leading zeros.
- $1 \leq L < 2^{100\ 001}$
Input
Input is given from Standard Input in the following format:
$L$
Output
Print the number of pairs $(a, b)$ that satisfy the conditions, modulo $10^9 + 7$.
Sample Input 1
10
Sample Output 1
5
Five pairs $(a, b)$ satisfy the conditions: $(0, 0), (0, 1), (1, 0), (0, 2)$ and $(2, 0)$.
Sample Input 2
1111111111111111111
Sample Output 2
162261460