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 Lua (5.3.2) (judge verdict: Accepted)

local read = setmetatable({}, {__index = function(t, k) local a = {} for i=1,#k do table.insert(a, '*'..string.sub(k, i, i)) end local r = io.read local u = table.unpack or unpack return function() return r(u(a)) end end})
read.N = function(N) local t={} for i=1,N do t[i]=read.n() end return t end
string.totable = function(s) local t={} local u=string.sub for i=1,#s do t[i] = u(s, i, i) end return t end
string.split = function(s) local t={} for w in string.gmatch(s, "[^%s]+") do table.insert(t, w) end return (table.unpack or unpack)(t) end
local function array(dimension, default_val) local n=dimension local m={}if default_val~=nil then m[1]={__index=function()return default_val end}end for i=2,n do m[i]={__index=function(p, k)local c=setmetatable({},m[i-1])rawset(p,k,c)return c end}end return setmetatable({},m[n])end
----
local MOD = math.floor(10^9)+7

local tmp = read.l():totable()
local S = {}
for i=1,#tmp do
    if tmp[i] == '?' then
        S[i] = '?'
    else
        S[i] = tonumber(tmp[i])
    end
end

local map_newm = {}
local map_oldm = {}
for nd=0,9 do
    for oldm=0,12 do
        local x = oldm * 10 + nd
        local y = (oldm * 10 + nd) % 13
        --print(nd, oldm, x, y)
        map_newm[x] = y
        map_oldm[x] = oldm
    end
end
local N = #S
local tbl = array(2, 0)
tbl[0][0] = 1
for i=1,N do
    local tio = tbl[i-1]
    local ti = tbl[i]
    if S[i] == '?' then
        for x=0,129 do
            local newm = map_newm[x]
            local oldm = map_oldm[x]
            ti[newm] = (ti[newm] + tio[oldm]) % MOD
        end
    else
        for oldm=0,12 do
            local newm = (oldm * 10 + S[i]) % 13
            ti[newm] = tio[oldm]
        end
    end
end
print(tbl[N][5])