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 OCaml (4.02.3) (judge verdict: Accepted)

let () =
    let main () =
        let m = 1_000_000_000 + 7 in
        let (+@) a b = (a + b) mod m in
        let s = read_line () in
        let a = Array.make 13 0 in
        let l = String.length s in
        let rec loop i a =
            if i = l then a else
                let b = Array.make 13 0 in
                let () =
                    if s.[i] = '?' then (
                        for j = 0 to 12 do
                            for k = 0 to 9 do
                                b.((j * 10 + k) mod 13) <- b.((j * 10 + k) mod 13) +@ a.(j)
                            done
                        done
                    ) else (
                        let k = int_of_char s.[i] - int_of_char '0' in
                        for j = 0 to 12 do
                            b.((j * 10 + k) mod 13) <- b.((j * 10 + k) mod 13) +@ a.(j)
                        done
                    )
                in
                loop (i + 1) b
        in
        let a = loop 0 (Array.init 13 (fun i -> if i = 0 then 1 else 0)) in
        Printf.printf "%d\n" a.(5)
    in
    main ()