XOR Triangle
Description
You are given a positive integer $n$. Since $n$ may be very large, you are given its binary representation.
You should compute the number of triples $(a,b,c)$ with $0 \leq a,b,c \leq n$ such that $a \oplus b$, $b \oplus c$, and $a \oplus c$ are the sides of a non-degenerate triangle.
Here, $\oplus$ denotes the bitwise XOR operation.
You should output the answer modulo $998\,244\,353$.
Three positive values $x$, $y$, and $z$ are the sides of a non-degenerate triangle if and only if $x+y>z$, $x+z>y$, and $y+z>x$.
The first and only line contains the binary representation of an integer $n$ ($0 < n < 2^{200\,000}$) without leading zeros.
For example, the string 10 is the binary representation of the number $2$, while the string 1010 represents the number $10$.
Print one integer — the number of triples $(a,b,c)$ satisfying the conditions described in the statement modulo $998\,244\,353$.
Input
The first and only line contains the binary representation of an integer $n$ ($0 < n < 2^{200\,000}$) without leading zeros.
For example, the string 10 is the binary representation of the number $2$, while the string 1010 represents the number $10$.
Output
Print one integer — the number of triples $(a,b,c)$ satisfying the conditions described in the statement modulo $998\,244\,353$.
Samples
101
12
1110
780
11011111101010010
141427753
Note
In the first test case, $101_2=5$.
- The triple $(a, b, c) = (0, 3, 5)$ is valid because $(a\oplus b, b\oplus c, c\oplus a) = (3, 6, 5)$ are the sides of a non-degenerate triangle.
- The triple $(a, b, c) = (1, 2, 4)$ is valid because $(a\oplus b, b\oplus c, c\oplus a) = (3, 6, 5)$ are the sides of a non-degenerate triangle.
The $6$ permutations of each of these two triples are all the valid triples, thus the answer is $12$.
In the third test case, $11\,011\,111\,101\,010\,010_2=114\,514$. The full answer (before taking the modulo) is $1\,466\,408\,118\,808\,164$.