Buggy python submission (verdict: Time Limit Exceeded):
import sys
sys.setrecursionlimit(1000000000)
import math
from fractions import gcd
from itertools import count, permutations
from functools import lru_cache
from collections import deque, defaultdict
from pprint import pprint
ii = lambda: int(input())
mis = lambda: map(int, input().split())
lmis = lambda: list(mis())
INF = float('inf')
N1097 = 10**9 + 7

def meg(f, ok, ng):
    while abs(ok-ng)>1:
        mid = (ok+ng)//2
        if f(mid):
            ok=mid
        else:
            ng=mid
    return ok
#

def main():
    S = input()
    d = 1
    dp = [0] * 13
    dp[0] = 1
    dp_n = [None] * 13
    for s in reversed(S):
        for i in range(13):
            dp[i] %= N1097
            dp_n[i] = 0
        if s == '?':
            for s in range(10):
                diff = (s*d) % 13
                for i in range(13):
                    dp_n[(i+diff)%13] += dp[i]
        else:
            s = int(s)
            diff = (s*d) % 13
            for i in range(13):
                dp_n[(i+diff)%13] += dp[i]
        dp_n, dp = dp, dp_n
        d = (10 * d) % 13
    print(dp[5] % N1097)

main()


Fixed submission:
import sys
sys.setrecursionlimit(1000000000)
import math
from fractions import gcd
from itertools import count, permutations
from functools import lru_cache
from collections import deque, defaultdict
from pprint import pprint
ii = lambda: int(input())
mis = lambda: map(int, input().split())
lmis = lambda: list(mis())
INF = float('inf')
N1097 = 10**9 + 7

def meg(f, ok, ng):
    while abs(ok-ng)>1:
        mid = (ok+ng)//2
        if f(mid):
            ok=mid
        else:
            ng=mid
    return ok
#

def main():
    S = input()
    d = 1
    dp = [0] * 13
    dp[0] = 1
    dp_n = [None] * 13
    for s in reversed(S):
        for i in range(13):
            dp[i] %= N1097
            dp_n[i] = 0
        if s == '?':
            for s in range(10):
                diff = (s*d) % 13
                for i in range(13):
                    dp_n[(i+diff)%13] += dp[i]
        else:
            s = int(s)
            diff = (s*d) % 13
            for i in range(13):
                dp_n[(i+diff)%13] += dp[i]
        dp_n, dp = dp, dp_n
        d = (10 * d) % 13
    print(dp[5] % N1097)

main()

Buggy python submission (verdict: Time Limit Exceeded):
S = input()[::-1]

a = [0] * 13
n = [0] * 13

from functools import lru_cache

@lru_cache(maxsize=10000000)
def get_base(d):
    if d==0:
        return 1
    base = get_base(d-1)
    base *= 10
    base %= 13
    return base

@lru_cache(maxsize=1000)
def get_shift(d, num):
    shift = num
    shift *= get_base(d)
    shift %= 13
    return shift

def calc(d, num):
#    shift = (num * (10 ** d))%13
    shift = get_shift(d, num)
    if d==0:
        n[shift] = 1
    else:
        for i in range(13):
            n[(i+shift)%13] += a[i]
    

for d, s in enumerate(S):
    if s=='?':
        for i in range(10):
            calc(d, i)
    else:
        calc(d, int(s[0]))
    a, n = n, a
    for i in range(13):
        a[i] %= 1000000007
        n[i] = 0

print(a[5])





Fixed submission:
import sys
sys.setrecursionlimit(1000000000)
import math
from fractions import gcd
from itertools import count, permutations
from functools import lru_cache
from collections import deque, defaultdict
from pprint import pprint
ii = lambda: int(input())
mis = lambda: map(int, input().split())
lmis = lambda: list(mis())
INF = float('inf')
N1097 = 10**9 + 7

def meg(f, ok, ng):
    while abs(ok-ng)>1:
        mid = (ok+ng)//2
        if f(mid):
            ok=mid
        else:
            ng=mid
    return ok
#

def main():
    S = input()
    d = 1
    dp = [0] * 13
    dp[0] = 1
    dp_n = [None] * 13
    for s in reversed(S):
        for i in range(13):
            dp[i] %= N1097
            dp_n[i] = 0
        if s == '?':
            for s in range(10):
                diff = (s*d) % 13
                for i in range(13):
                    dp_n[(i+diff)%13] += dp[i]
        else:
            s = int(s)
            diff = (s*d) % 13
            for i in range(13):
                dp_n[(i+diff)%13] += dp[i]
        dp_n, dp = dp, dp_n
        d = (10 * d) % 13
    print(dp[5] % N1097)

main()

Buggy python submission (verdict: Time Limit Exceeded):
S = input()[::-1]

a = [0] * 13
n = [0] * 13

from functools import lru_cache

@lru_cache(maxsize=10000000)
def get_base(d):
    base = 1
    for _ in range(d):
        base *= 10
        base %= 13
    return base

@lru_cache(maxsize=1000)
def get_shift(d, num):
    shift = num
    shift *= get_base(d)
    shift %= 13
    return shift

def calc(d, num):
#    shift = (num * (10 ** d))%13
    shift = get_shift(d, num)
    if d==0:
        n[shift] = 1
    else:
        for i in range(13):
            n[(i+shift)%13] += a[i]
    

for d, s in enumerate(S):
    if s=='?':
        for i in range(10):
            calc(d, i)
    else:
        calc(d, int(s[0]))
    a, n = n, a
    for i in range(13):
        a[i] %= 1000000007
        n[i] = 0

print(a[5])





Fixed submission:
import sys
sys.setrecursionlimit(1000000000)
import math
from fractions import gcd
from itertools import count, permutations
from functools import lru_cache
from collections import deque, defaultdict
from pprint import pprint
ii = lambda: int(input())
mis = lambda: map(int, input().split())
lmis = lambda: list(mis())
INF = float('inf')
N1097 = 10**9 + 7

def meg(f, ok, ng):
    while abs(ok-ng)>1:
        mid = (ok+ng)//2
        if f(mid):
            ok=mid
        else:
            ng=mid
    return ok
#

def main():
    S = input()
    d = 1
    dp = [0] * 13
    dp[0] = 1
    dp_n = [None] * 13
    for s in reversed(S):
        for i in range(13):
            dp[i] %= N1097
            dp_n[i] = 0
        if s == '?':
            for s in range(10):
                diff = (s*d) % 13
                for i in range(13):
                    dp_n[(i+diff)%13] += dp[i]
        else:
            s = int(s)
            diff = (s*d) % 13
            for i in range(13):
                dp_n[(i+diff)%13] += dp[i]
        dp_n, dp = dp, dp_n
        d = (10 * d) % 13
    print(dp[5] % N1097)

main()

Buggy python submission (verdict: Time Limit Exceeded):
S = input()[::-1]

a = [0] * 13
n = [0] * 13

def calc(d, num):
#    shift = (num * (10 ** d))%13
    shift = num
    for _ in range(d):
        shift *= 10
        shift %= 13
    if d==0:
        n[shift] = 1
    else:
        for i in range(13):
            n[(i+shift)%13] += a[i]
    

for d, s in enumerate(S):
    if s=='?':
        for i in range(10):
            calc(d, i)
    else:
        calc(d, int(s[0]))
    a, n = n, a
    for i in range(13):
        a[i] %= 1000000007
        n[i] = 0

print(a[5])





Fixed submission:
import sys
sys.setrecursionlimit(1000000000)
import math
from fractions import gcd
from itertools import count, permutations
from functools import lru_cache
from collections import deque, defaultdict
from pprint import pprint
ii = lambda: int(input())
mis = lambda: map(int, input().split())
lmis = lambda: list(mis())
INF = float('inf')
N1097 = 10**9 + 7

def meg(f, ok, ng):
    while abs(ok-ng)>1:
        mid = (ok+ng)//2
        if f(mid):
            ok=mid
        else:
            ng=mid
    return ok
#

def main():
    S = input()
    d = 1
    dp = [0] * 13
    dp[0] = 1
    dp_n = [None] * 13
    for s in reversed(S):
        for i in range(13):
            dp[i] %= N1097
            dp_n[i] = 0
        if s == '?':
            for s in range(10):
                diff = (s*d) % 13
                for i in range(13):
                    dp_n[(i+diff)%13] += dp[i]
        else:
            s = int(s)
            diff = (s*d) % 13
            for i in range(13):
                dp_n[(i+diff)%13] += dp[i]
        dp_n, dp = dp, dp_n
        d = (10 * d) % 13
    print(dp[5] % N1097)

main()

Buggy python submission (verdict: Time Limit Exceeded):
S = input()[::-1]

a = [0] * 13
n = [0] * 13

def calc(d, num):
    shift = (num * (10 ** d))%13
    if d==0:
        n[shift] = 1
    else:
        for i in range(13):
            n[(i+shift)%13] += a[i]
    

for d, s in enumerate(S):
    if s=='?':
        for i in range(10):
            calc(d, i)
    else:
        calc(d, int(s[0]))
    a, n = n, a
    for i in range(13):
        a[i] %= 1000000007
        n[i] = 0

print(a[5])





Fixed submission:
import sys
sys.setrecursionlimit(1000000000)
import math
from fractions import gcd
from itertools import count, permutations
from functools import lru_cache
from collections import deque, defaultdict
from pprint import pprint
ii = lambda: int(input())
mis = lambda: map(int, input().split())
lmis = lambda: list(mis())
INF = float('inf')
N1097 = 10**9 + 7

def meg(f, ok, ng):
    while abs(ok-ng)>1:
        mid = (ok+ng)//2
        if f(mid):
            ok=mid
        else:
            ng=mid
    return ok
#

def main():
    S = input()
    d = 1
    dp = [0] * 13
    dp[0] = 1
    dp_n = [None] * 13
    for s in reversed(S):
        for i in range(13):
            dp[i] %= N1097
            dp_n[i] = 0
        if s == '?':
            for s in range(10):
                diff = (s*d) % 13
                for i in range(13):
                    dp_n[(i+diff)%13] += dp[i]
        else:
            s = int(s)
            diff = (s*d) % 13
            for i in range(13):
                dp_n[(i+diff)%13] += dp[i]
        dp_n, dp = dp, dp_n
        d = (10 * d) % 13
    print(dp[5] % N1097)

main()