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C. XOR Triangle
time limit per test
4 seconds
memory limit per test
512 megabytes
input
standard input
output
standard output

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 $y+z>x$.

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 $998\,244\,353$.

Examples
Input
101
Output
12

Input
1110
Output
780

Input
11011111101010010
Output
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 $1\,466\,408\,118\,808\,164$.