Problem:
Score : 400 points

Problem Statement

Given is a string S. Each character in S is either a digit (0, ..., 9) or ?.

Among the integers obtained by replacing each occurrence of ? with a digit, how many have a remainder of 5 when divided by 13? An integer may begin with 0.

Since the answer can be enormous, print the count modulo 10^9+7.

Constraints

S is a string consisting of digits (0, ..., 9) and ?.

1 \leq |S| \leq 10^5

Input

Input is given from Standard Input in the following format:

S

Output

Print the number of integers satisfying the condition, modulo 10^9+7.

Sample Input 1

??2??5

Sample Output 1

768

For example, 482305, 002865, and 972665 satisfy the condition.

Sample Input 2

?44

Sample Output 2

1

Only 044 satisfies the condition.

Sample Input 3

7?4

Sample Output 3

0

We may not be able to produce an integer satisfying the condition.

Sample Input 4

?6?42???8??2??06243????9??3???7258??5??7???????774????4?1??17???9?5?70???76???

Sample Output 4

153716888

Solution in Rust (1.42.0) (judge verdict: Accepted)

#![allow(unused_imports)]
#![allow(non_snake_case)]
use std::cmp::*;
use std::collections::*;
use std::io::Write;

#[allow(unused_macros)]
macro_rules! debug {
    ($($e:expr),*) => {
        #[cfg(debug_assertions)]
        $({
            let (e, mut err) = (stringify!($e), std::io::stderr());
            writeln!(err, "{} = {:?}", e, $e).unwrap()
        })*
    };
}

fn main() {
    let s = read::<String>()
        .chars()
        .map(|x| x.to_digit(10))
        .collect::<Vec<_>>();
    let mut dp = vec![Modulo(0); 13];
    dp[0] = Modulo(1);
    for i in 0..s.len() {
        let mut dp1 = vec![Modulo(0); 13];
        for prev in 0..13 {
            let next = (prev * 10) % 13;
            dp1[next] += dp[prev];
        }
        let mut dp2 = vec![Modulo(0); 13];
        if let Some(x) = s[i] {
            for prev in 0..13 {
                let next = (prev + x as usize) % 13;
                dp2[next] += dp1[prev];
            }
        } else {
            for xi in 0..10 {
                for prev in 0..13 {
                    let next = (prev + xi) % 13;
                    dp2[next] += dp1[prev];
                }
            }
        }
        dp = dp2;
    }
    println!("{}", dp[5]);
}

#[derive(Clone, Copy, Debug, Default, Eq, Hash, Ord, PartialEq, PartialOrd)]
struct Modulo(i64);
static mut MODULUS: i64 = 1000_000_000 + 7;
impl Modulo {
    fn set_modulus(m: i64) {
        unsafe {
            MODULUS = m;
        }
    }
    fn get_modulus() -> i64 {
        unsafe { MODULUS }
    }
    fn new(x: i64) -> Modulo {
        let m = Modulo::get_modulus();
        if x < 0 {
            Modulo(x % m + m)
        } else if x < m {
            Modulo(x)
        } else {
            Modulo(x % m)
        }
    }
    fn pow(self, p: i64) -> Modulo {
        if p == 0 {
            Modulo(1)
        } else {
            let mut t = self.pow(p / 2);
            t *= t;
            if p & 1 == 1 {
                t *= self;
            }
            t
        }
    }
    fn inv(self) -> Modulo {
        self.pow(Modulo::get_modulus() - 2)
    }
}
impl std::fmt::Display for Modulo {
    fn fmt(&self, f: &mut std::fmt::Formatter) -> std::fmt::Result {
        self.0.fmt(f)
    }
}
impl std::ops::AddAssign for Modulo {
    fn add_assign(&mut self, other: Modulo) {
        let m = Modulo::get_modulus();
        self.0 += other.0;
        if self.0 >= m {
            self.0 -= m;
        }
    }
}
impl std::ops::MulAssign for Modulo {
    fn mul_assign(&mut self, other: Modulo) {
        let m = Modulo::get_modulus();
        self.0 *= other.0;
        self.0 %= m;
    }
}
impl std::ops::SubAssign for Modulo {
    fn sub_assign(&mut self, other: Modulo) {
        let m = Modulo::get_modulus();
        self.0 += m - other.0;
        if self.0 >= m {
            self.0 -= m;
        }
    }
}
macro_rules! impl_modulo_ops {
    ($imp:ident, $method:ident, $assign_imp:ident, $assign_method:ident) => {
        impl<'a> std::ops::$assign_imp<&'a Modulo> for Modulo {
            fn $assign_method(&mut self, other: &'a Modulo) {
                std::ops::$assign_imp::$assign_method(self, *other);
            }
        }
        impl std::ops::$imp for Modulo {
            type Output = Modulo;
            fn $method(self, other: Modulo) -> Modulo {
                let mut x = self;
                std::ops::$assign_imp::$assign_method(&mut x, other);
                x
            }
        }
        impl<'a> std::ops::$imp<Modulo> for &'a Modulo {
            type Output = Modulo;
            fn $method(self, other: Modulo) -> Modulo {
                std::ops::$imp::$method(*self, other)
            }
        }
        impl<'a> std::ops::$imp<&'a Modulo> for Modulo {
            type Output = Modulo;
            fn $method(self, other: &'a Modulo) -> Modulo {
                std::ops::$imp::$method(self, *other)
            }
        }
        impl<'a, 'b> std::ops::$imp<&'b Modulo> for &'a Modulo {
            type Output = Modulo;
            fn $method(self, other: &'b Modulo) -> Modulo {
                std::ops::$imp::$method(*self, *other)
            }
        }
        impl std::ops::$assign_imp<i64> for Modulo {
            fn $assign_method(&mut self, other: i64) {
                std::ops::$assign_imp::$assign_method(self, Modulo::new(other));
            }
        }
        impl<'a> std::ops::$assign_imp<&'a i64> for Modulo {
            fn $assign_method(&mut self, other: &'a i64) {
                std::ops::$assign_imp::$assign_method(self, *other);
            }
        }
        impl std::ops::$imp<i64> for Modulo {
            type Output = Modulo;
            fn $method(self, other: i64) -> Modulo {
                let mut x = self;
                std::ops::$assign_imp::$assign_method(&mut x, other);
                x
            }
        }
        impl<'a> std::ops::$imp<&'a i64> for Modulo {
            type Output = Modulo;
            fn $method(self, other: &'a i64) -> Modulo {
                std::ops::$imp::$method(self, *other)
            }
        }
        impl<'a> std::ops::$imp<i64> for &'a Modulo {
            type Output = Modulo;
            fn $method(self, other: i64) -> Modulo {
                std::ops::$imp::$method(*self, other)
            }
        }
        impl<'a, 'b> std::ops::$imp<&'b i64> for &'a Modulo {
            type Output = Modulo;
            fn $method(self, other: &'b i64) -> Modulo {
                std::ops::$imp::$method(*self, *other)
            }
        }
    };
}
impl_modulo_ops!(Add, add, AddAssign, add_assign);
impl_modulo_ops!(Mul, mul, MulAssign, mul_assign);
impl_modulo_ops!(Sub, sub, SubAssign, sub_assign);

use std::iter::Sum;
impl Sum for Modulo {
    fn sum<I>(iter: I) -> Self
    where
        I: Iterator<Item = Modulo>,
    {
        iter.fold(Modulo(0), |a, b| a + b)
    }
}

impl<'a> Sum<&'a Modulo> for Modulo {
    fn sum<I>(iter: I) -> Self
    where
        I: Iterator<Item = &'a Self>,
    {
        iter.fold(Modulo(0), |a, b| a + b)
    }
}

fn read<T: std::str::FromStr>() -> T {
    let mut s = String::new();
    std::io::stdin().read_line(&mut s).ok();
    s.trim().parse().ok().unwrap()
}

fn read_vec<T: std::str::FromStr>() -> Vec<T> {
    read::<String>()
        .split_whitespace()
        .map(|e| e.parse().ok().unwrap())
        .collect()
}