Harryxun/testdata huggingface dataset:
Buggy python submission (verdict: Runtime Error):
import numpy as np
N, P = map(int,input().split())
S = list(map(int,input()))

ans = 0
if P == 2:
    for i,s in enumerate(S):
        if s % 2 == 0:
            ans += i + 1
elif P == 5:
    for i,s in enumerate(S):
        if s % 5 == 0:
            ans += i + 1
else:
    res = []
    n = 0
    for s in S[::-1]:
        n = (10*n + s) % P
        res.append(n)

    res.append(0) # right

    inds, cnts = np.unique(res, return_counts=True)

    for cnt in cnts:
        if cnt == 1:
            continue
        ans += cnt * (cnt - 1) // 2
print(ans)

    



Fixed submission:
from collections import Counter
N, P = map(int,input().split())
S = input()

'''
N, P  = 10, 7
S = '1234567890'
tans = 0
for i in range(N):
    for j in range(i+1,N+1):
        t = int(S[i:j])
        tans += ((t % P) == 0)
print(tans)
'''

S = list(map(int,S))

ans = 0
if P == 2:
    for i,s in enumerate(S):
        if s % 2 == 0:
            ans += i + 1
elif P == 5:
    for i,s in enumerate(S):
        if s % 5 == 0:
            ans += i + 1
else:
    res = []
    n = 0
    k = 1
    for s in S[::-1]:
        n = (s * k + n) % P
        k = (k * 10) % P
        res.append(n)

    res.append(0) # right
    #print(N,P,S)
    #print(res)

    cnts = Counter(res)
    for ind, cnt in cnts.items():
        if cnt == 1:
            continue
        ans += cnt * (cnt - 1) // 2

print(ans)

Buggy python submission (verdict: Time Limit Exceeded):
from collections import defaultdict
N, P = map(int,input().split())
S = input()

dp = [defaultdict(int) for i in range(N)]


dp[0][int(S[0]) % P] = 1

for i in range(N-1):
    n = int(S[i+1])
    #print(n)
    for k,v in dp[i].items():
        dp[i+1][(10*k+n) % P] += v
    dp[i+1][n % P] += 1
    #print(dp[i+1])

ans = sum([dp[i][0] for i in range(N)])
print(ans)




Fixed submission:
from collections import Counter
N, P = map(int,input().split())
S = input()

'''
N, P  = 10, 7
S = '1234567890'
tans = 0
for i in range(N):
    for j in range(i+1,N+1):
        t = int(S[i:j])
        tans += ((t % P) == 0)
print(tans)
'''

S = list(map(int,S))

ans = 0
if P == 2:
    for i,s in enumerate(S):
        if s % 2 == 0:
            ans += i + 1
elif P == 5:
    for i,s in enumerate(S):
        if s % 5 == 0:
            ans += i + 1
else:
    res = []
    n = 0
    k = 1
    for s in S[::-1]:
        n = (s * k + n) % P
        k = (k * 10) % P
        res.append(n)

    res.append(0) # right
    #print(N,P,S)
    #print(res)

    cnts = Counter(res)
    for ind, cnt in cnts.items():
        if cnt == 1:
            continue
        ans += cnt * (cnt - 1) // 2

print(ans)
Buggy python submission (verdict: Time Limit Exceeded):
from collections import defaultdict, deque, Counter
N,M,K= map(int,input().split())

friend = defaultdict(list)
#block = defaultdict(list)

for i in range(M):
    a,b= list(map(int,input().split()))
    friend[a].append(b)
    friend[b].append(a)
'''
for i in range(K):
    c,d= list(map(int,input().split()))
    block[c].append(d)
    block[d].append(c)
    '''
block = [list(map(int,input().split())) for i in range(K)]

IDs = [-1] * (N+1)
ID = 0
for i in range(1,N+1):
    if IDs[i] != -1:
        continue
    dq = deque([i])
    while dq:
        v = dq.popleft()
        IDs[v] = ID
        for nv in friend[v]:
            if IDs[nv] == -1:
                dq.append(nv)
    ID += 1

group = Counter(IDs)

ans = []
for i in range(1,N+1):
    temp = group[IDs[i]] - len(friend[i]) - 1
    ans.append(temp)

for c,d in block:
    if IDs[c] == IDs[d]:
        ans[c-1] -= 1
        ans[d-1] -= 1

print(*ans)









Fixed submission:
from collections import defaultdict, deque, Counter
N,M,K= map(int,input().split())

friend = defaultdict(list)
#block = defaultdict(list)

for i in range(M):
    a,b= list(map(int,input().split()))
    friend[a].append(b)
    friend[b].append(a)
'''
for i in range(K):
    c,d= list(map(int,input().split()))
    block[c].append(d)
    block[d].append(c)
    '''
block = [list(map(int,input().split())) for i in range(K)]

IDs = [-1] * (N+1)
ID = 0
for i in range(1,N+1):
    if IDs[i] != -1:
        continue
    dq = deque([i])
    while dq:
        v = dq.popleft()
        IDs[v] = ID
        for nv in friend[v]:
            if IDs[nv] == -1:
                dq.append(nv)
    ID += 1

group = Counter(IDs)

ans = []
for i in range(1,N+1):
    temp = group[IDs[i]] - len(friend[i]) - 1
    ans.append(temp)

for c,d in block:
    if IDs[c] == IDs[d]:
        ans[c-1] -= 1
        ans[d-1] -= 1

print(*ans)
Buggy python submission (verdict: Time Limit Exceeded):
import sys
sys.setrecursionlimit(10**8)
N = int(input())

memo = {0:0,1:1}
def f(price):
    if price in memo:
        return memo[price]
    n = price % 10
    res = min(f(price//10 + 1) + 10-n, f(price//10) + n)
    memo[price] = res
    return res

print(f(N))


Fixed submission:
S = input()
N = list(map(int,S))
L = len(N)

dp = [[0]*2 for i in [0]*(L+1)]
dp[0][1] = 1

for i in range(L):
    n = N[i]
    dp[i+1][0] = min(dp[i][0] + n, dp[i][1] + 10-n)
    if n == 9:
        dp[i+1][1] = dp[i][1]
    else:
        dp[i+1][1] = min(dp[i][0] + n + 1, dp[i][1] + 10 - (n+1))

print(dp[L][0])
Buggy python submission (verdict: Time Limit Exceeded):
MOD = 10**9 + 7
r1,c1,r2,c2 = map(int,input().split())

n = 2000100
fact = [1]*(n+1)
inv = [1]*(n+1)
ifact = [1]*(n+1)
for i in range(1,n+1):
    fact[i] = fact[i-1]*i % MOD
    inv[i] = pow(i,MOD-2,MOD)
    ifact[i] = ifact[i-1]*inv[i] % MOD

def nCr(n,r):
    return fact[n] * ifact[r] * ifact[n-r] % MOD

def f(r,c):
    temp = [nCr(k,c) for k in range(c+1,r+c+2)]
    return sum(temp) % MOD
    
print((f(r2,c2) - f(r2,c1-1) - f(r1-1,c2) + f(r1-1,c1-1)) % MOD)


Fixed submission:
MOD = 10**9 + 7
r1,c1,r2,c2 = map(int,input().split())

n = 2000100
fact = [1]*(n+1)
inv = [1]*(n+1)
ifact = [1]*(n+1)
for i in range(1,n+1):
    fact[i] = fact[i-1]*i % MOD
    inv[i] = pow(i,MOD-2,MOD)
    ifact[i] = ifact[i-1]*inv[i] % MOD

def nCr(n,r):
    return fact[n] * ifact[r] * ifact[n-r] % MOD

def f(r,c):
    return nCr(r+c+2,c+1)-1
    
print((f(r2,c2) - f(r2,c1-1) - f(r1-1,c2) + f(r1-1,c1-1)) % MOD)

Buggy python submission (verdict: Runtime Error):
MOD = 10**9 + 7
r1,c1,r2,c2 = map(int,input().split())

# f(r,c) = (r+c)_C_c

n = 200100
fact = [1]*(n+1)
inv = [1]*(n+1)
ifact = [1]*(n+1)
for i in range(1,n+1):
    fact[i] = fact[i-1]*i % MOD
    inv[i] = pow(i,MOD-2,MOD)
    ifact[i] = ifact[i-1]*inv[i] % MOD

def nCr(n,r):
    return fact[n] * ifact[r] * ifact[n-r] % MOD

def f(r,c):
    temp = [nCr(k,c) for k in range(c+1,r+c+2)]
    return sum(temp) % MOD
    
print((f(r2,c2) - f(r2,c1-1) - f(r1-1,c2) + f(r1-1,c1-1)) % MOD)


Fixed submission:
MOD = 10**9 + 7
r1,c1,r2,c2 = map(int,input().split())

n = 2000100
fact = [1]*(n+1)
inv = [1]*(n+1)
ifact = [1]*(n+1)
for i in range(1,n+1):
    fact[i] = fact[i-1]*i % MOD
    inv[i] = pow(i,MOD-2,MOD)
    ifact[i] = ifact[i-1]*inv[i] % MOD

def nCr(n,r):
    return fact[n] * ifact[r] * ifact[n-r] % MOD

def f(r,c):
    return nCr(r+c+2,c+1)-1
    
print((f(r2,c2) - f(r2,c1-1) - f(r1-1,c2) + f(r1-1,c1-1)) % MOD)
Buggy python submission (verdict: Time Limit Exceeded):
from collections import defaultdict
H, N= map(int,input().split())
A = []
B = []
for i in range(N):
    a,b = map(int,input().split())
    A.append(a)
    B.append(b)

dp = defaultdict(lambda : 10**10)
dp[0] = 0

for i in range(0,N):
    ndp = dp.copy()
    a = A[i]
    b = B[i]
    for h,m in dp.items():
        cnt = 0
        while True:
            damage = h+a*cnt
            if damage >= H:
                ndp[H] = min(ndp[H], ndp[h] + b*cnt)
                break
            ndp[damage] = min(ndp[damage], ndp[h] + b*cnt)
            cnt += 1
    dp = ndp

print(dp[H])




Fixed submission:
from collections import defaultdict
H, N= map(int,input().split())
A = []
B = []
for i in range(N):
    a,b = map(int,input().split())
    A.append(a)
    B.append(b)

dp = [10**10]*(H+1)
dp[0] = 0

for i in range(0,N):
    a = A[i]
    b = B[i]
    for h in range(H+1-a):
        dp[h+a] = min(dp[h+a], dp[h] + b)
    dp[H] = min(dp[H], min(dp[H-a:H])+b)

print(dp[H])
Buggy python submission (verdict: Time Limit Exceeded):
import bisect
import sys
input = sys.stdin.buffer.readline
N,D,A= map(int,input().split())
XH = [list(map(int,input().split())) for i in range(N)]
XH.sort()
X,H = [list(x) for x in zip(*XH)]

ans = 0
for i in range(N):
    x = X[i]
    if H[i] <= 0:
        continue
    cnt_bom = -(-H[i] // A)
    ans += cnt_bom
    damage = cnt_bom * A
    ind = bisect.bisect_right(X,x+2*D)
    for j in range(i,ind):
        H[j] -= damage
print(ans)


Fixed submission:
import bisect
import sys
input = sys.stdin.buffer.readline
N,D,A= map(int,input().split())
XH = [list(map(int,input().split())) for i in range(N)]
XH.sort()
X,H = [list(x) for x in zip(*XH)]

ans = 0
cum = [0]*N
dam = 0
for i in range(N):
    dam += cum[i]
    H[i] -= dam
    x = X[i]
    if H[i] <= 0:
        continue
    cnt_bom = -(-H[i] // A)
    ans += cnt_bom
    damage = cnt_bom * A
    ind = bisect.bisect_right(X,x+2*D)
    if i+1 < N:
        cum[i+1] += damage 
    if ind < N:
        cum[ind] -=damage 

print(ans)

Buggy python submission (verdict: Time Limit Exceeded):
import bisect
import sys
input = sys.stdin.buffer.readline
N,D,A= map(int,input().split())
XH = [list(map(int,input().split())) for i in range(N)]
XH.sort()
X,H = [list(x) for x in zip(*XH)]

ans = 0
for i in range(N):
    x = X[i]
    if H[i] <= 0:
        continue
    cnt_bom = -(-H[i] // A)
    ans += cnt_bom
    damage = cnt_bom * A
    ind = bisect.bisect_right(X,x+2*D)
    for j in range(i,ind):
        H[j] -= damage
print(ans)


Fixed submission:
import bisect
import sys
input = sys.stdin.buffer.readline
N,D,A= map(int,input().split())
XH = [list(map(int,input().split())) for i in range(N)]
XH.sort()
X,H = [list(x) for x in zip(*XH)]

ans = 0
cum = [0]*N
dam = 0
for i in range(N):
    dam += cum[i]
    H[i] -= dam
    x = X[i]
    if H[i] <= 0:
        continue
    cnt_bom = -(-H[i] // A)
    ans += cnt_bom
    damage = cnt_bom * A
    ind = bisect.bisect_right(X,x+2*D)
    if i+1 < N:
        cum[i+1] += damage 
    if ind < N:
        cum[ind] -=damage 

print(ans)
Buggy python submission (verdict: Time Limit Exceeded):
import functools
MOD = 10**9 + 7
N = int(input())
A = list(map(int,input().split()))

def gcd(a,b):
    while b:
        a,b = b, a%b
    return a

def lcm(a,b):
    return (a*b//gcd(a,b))

def list_lcm(l):
    return functools.reduce(lcm,l)

g = list_lcm(A)

B = [g // a for a in A]
print(sum(B) % MOD)



Fixed submission:
import functools
MOD = 10**9 + 7
N = int(input())
A = list(map(int,input().split()))

def gcd(a,b):
    while b:
        a,b = b, a%b
    return a

def lcm(a,b):
    return (a*b // gcd(a,b))

def list_lcm(l):
    return functools.reduce(lcm,l)

g = list_lcm(A)

B = [g //a for a in A]
print(sum(B) % MOD)
Buggy python submission (verdict: Runtime Error):
MOD = 10**9 + 7
N,K= map(int,input().split())
A = list(map(int,input().split()))
A.sort()

if K == 1:
    print(0)
    exit()

n = 10000
fact = [1]*(n+1)
inv = [1]*(n+1)
ifact = [1]*(n+1)
for i in range(1,n+1):
    fact[i] = fact[i-1]*i % MOD
    inv[i] = pow(i,MOD-2,MOD)
    ifact[i] = ifact[i-1]*inv[i] % MOD

def nCr(n,r):
    return fact[n] * ifact[r] * ifact[n-r] % MOD

ans = 0
for l in range(N):
    for r in range(l+K-1,N):
        temp = (A[r]-A[l]) * nCr(r-l-1,K-2)
        ans = (ans + temp) % MOD

print(ans)






Fixed submission:
MOD = 10**9 + 7
N,K= map(int,input().split())
A = list(map(int,input().split()))
#N = 100000
#K = 10
#A = list(range(N))

A.sort()

if K == 1:
    print(0)
    exit()

n = 100001
fact = [1]*(n+1)
inv = [1]*(n+1)
ifact = [1]*(n+1)
for i in range(1,n+1):
    fact[i] = (fact[i-1]*i) % MOD
    inv[i] = pow(i,MOD-2,MOD)
    ifact[i] = (ifact[i-1]*inv[i]) % MOD

def nCr(n,r):
    return (fact[n] * ifact[r] * ifact[n-r]) % MOD

ans = 0

for i in range(K-1,N):
    ans += A[i] * nCr(i,K-1)
    ans %= MOD

for i in range(N-K+1):
    ans -= A[i] * nCr(N-i-1,K-1)
    ans %= MOD

'''
for l in range(N):
    for r in range(l+K-1,N):
        temp = (A[r]-A[l]) * nCr(r-l-1,K-2)
        ans = (ans + temp) % MOD
        '''

print(ans)
Buggy python submission (verdict: Time Limit Exceeded):
MOD = 10**9 + 7
N = int(input())
A = list(map(int,input().split()))

B = [bin(a)[2:].zfill(61)[::-1] for a in A]
N1 = []
for i in range(0,61):
    m = 0
    for b in B:
        if b[i] == "1":
            m += 1
    N1.append(m)

ans = 0
for i,n1 in enumerate(N1):
    n0 = N-n1
    m = n0 * n1
    ans += 2**i * m
    ans = ans % MOD
print(ans % MOD)






Fixed submission:
MOD = 10**9 + 7
N = int(input())
A = list(map(int,input().split()))

#B = [bin(a)[2:].zfill(61)[::-1] for a in A]
N1 = []
for i in range(0,61):
    m = sum([a % 2 for a in A])
    N1.append(m)
    for j in range(N):
        A[j] = A[j] >> 1

ans = 0
for i,n1 in enumerate(N1):
    n0 = N-n1
    m = n0 * n1
    ans += 2**i * m
    ans = ans % MOD
print(ans % MOD)
Buggy python submission (verdict: Time Limit Exceeded):
N,K= map(int,input().split())
A = list(map(int,input().split()))

ans = 0
T = min(N,K)
for l in range(1,T+1):
    f = sum(A[:l])
    if f % K == l:
        ans += 1
    for j in range(l,N):
        f = f - A[j-l] + A[j]
        if f % K == l:
            ans += 1
print(ans)



Fixed submission:
import numpy as np
from collections import defaultdict,deque
N,K= map(int,input().split())
A = np.array([1]+list(map(int,input().split())))-1

Acs = np.cumsum(A)
Acs %= K

ans = 0
cnt = defaultdict(deque)
for i,c in enumerate(Acs):
    cnt[c].append(i)
    while cnt[c]:
        if i-cnt[c][0]> K-1:
            cnt[c].popleft()
        else:
            break
    ans += len(cnt[c]) - 1
print(ans)
Buggy python submission (verdict: Runtime Error):
N,K= map(int,input().split())
H= list(map(int,input().split()))

def dpinit(ps, val=0):
    res = [val for i in [0]*ps[-1]]
    for i in ps[:-1][::-1]:
        res = [res.copy() for k in [0]*i]
    return res

H = [0] + H + [0]
dp = dpinit((N+2,K+2))

for i in range(1,N+2):
    for j in range(K+1):
        l = min(j+1,i)
        cand = [dp[i-k][j-k+1] + max(0, H[i]-H[i-k]) for k in range(1,l+1)]
        dp[i][j] = min(cand)

print(dp[N+1][K])



Fixed submission:
N,K= map(int,input().split())
H= list(map(int,input().split()))

def dpinit(ps, val=0):
    res = [val for i in [0]*ps[-1]]
    for i in ps[:-1][::-1]:
        res = [res[:] for k in [0]*i]
    return res

H = [0] + H + [0]
dp = dpinit((N+2,K+2))

for i in range(1,N+2):
    for j in range(K+1):
        l = min(j+1,i)
        cand = [dp[i-k][j-k+1] + max(0, H[i]-H[i-k]) for k in range(1,l+1)]
        dp[i][j] = min(cand)

print(dp[N+1][K])

Buggy python submission (verdict: Time Limit Exceeded):
N,K= map(int,input().split())
H= list(map(int,input().split()))

def dpinit(ps, val=0):
    res = [val for i in [0]*ps[-1]]
    for i in ps[:-1][::-1]:
        res = [res.copy() for k in [0]*i]
    return res

H = [0] + H + [0]
dp = dpinit((N+2,K+2))

for i in range(1,N+2):
    for j in range(K+1):
        l = min(j+1,i)
        cand = [dp[i-k][j-k+1] + max(0, H[i]-H[i-k]) for k in range(1,l+1)]
        dp[i][j] = min(cand)

print(dp[N+1][K])



Fixed submission:
N,K= map(int,input().split())
H= list(map(int,input().split()))

def dpinit(ps, val=0):
    res = [val for i in [0]*ps[-1]]
    for i in ps[:-1][::-1]:
        res = [res[:] for k in [0]*i]
    return res

H = [0] + H + [0]
dp = dpinit((N+2,K+2))

for i in range(1,N+2):
    for j in range(K+1):
        l = min(j+1,i)
        cand = [dp[i-k][j-k+1] + max(0, H[i]-H[i-k]) for k in range(1,l+1)]
        dp[i][j] = min(cand)

print(dp[N+1][K])

Buggy python submission (verdict: Runtime Error):
from time import time

N,K= map(int,input().split())
H= list(map(int,input().split()))

def dpinit(ps, val=0):
    res = [val for i in [0]*ps[-1]]
    for i in ps[:-1][::-1]:
        res = [res.copy() for k in [0]*i]
    return res

H = [0] + H + [0]
dp = dpinit((N+2,K+2))

for i in range(1,N+2):
    for j in range(K+1):
        l = min(j+1,i)
        cand = [dp[i-k][j-k+1] + max(0, H[i]-H[i-k]) for k in range(1,l+1)]
        dp[i][j] = min(cand)

print(dp[N+1][K])



Fixed submission:
N,K= map(int,input().split())
H= list(map(int,input().split()))

def dpinit(ps, val=0):
    res = [val for i in [0]*ps[-1]]
    for i in ps[:-1][::-1]:
        res = [res[:] for k in [0]*i]
    return res

H = [0] + H + [0]
dp = dpinit((N+2,K+2))

for i in range(1,N+2):
    for j in range(K+1):
        l = min(j+1,i)
        cand = [dp[i-k][j-k+1] + max(0, H[i]-H[i-k]) for k in range(1,l+1)]
        dp[i][j] = min(cand)

print(dp[N+1][K])
Buggy python submission (verdict: Time Limit Exceeded):
import bisect
N = int(input())
P = list(map(int,input().split(' ')))

ls = [-1,-1,N,N]
ans = 0
for i in range(N,0,-1):
    x = P.index(i)
    ind = bisect.bisect(ls,x)
    bisect.insort(ls,x)
    L2 = ls[ind-2]
    L1 = ls[ind-1]
    R1 = ls[ind+1]
    R2 = ls[ind+2]
    L = (L1-L2)*(R1-x)
    R = (R2-R1)*(x-L1)
    ans += i * (L+R)
print(ans)


Fixed submission:
import sys
input = sys.stdin.readline
N = int(input())
P = list(map(int,input().split()))

'''
import random
N = 100000
P = list(range(1,N+1))
#random.shuffle(P)
'''

class SegmentTree:
    def __init__(self,data,op,default):
        N = len(data)
        self.N = N
        self.op = op
        self.default = default
        self.l = 2**((N-1).bit_length())
        self.data = [default]*self.l + data + [default]*(self.l-N)
        for i in range(self.l-1,0,-1):
            self.data[i] = op(self.data[2*i], self.data[2*i+1])

    def get_data(self):
        return self.data[self.l:self.l + self.N]

    def set(self,i,val):
        i += self.l
        self.data[i] = val
        i = i//2
        while i > 0:
            self.data[i] = self.op(self.data[2*i], self.data[2*i+1])
            i = i//2

    def get(self,i,j):
        i += self.l
        j += self.l
        s = self.default 
        while j-i > 0:
            if i & 1:
                s = self.op(s,self.data[i])
                i += 1
            if j & 1:
                s = self.op(s,self.data[j-1])
                j -= 1
            i = i//2
            j = j//2
        return s

P2 = [[[p,i],[0,0]] for i,p in enumerate(P)]

max2 = lambda x,y : sorted(x+y,reverse=True)[:2]

seg = SegmentTree(P2,max2,[[0,0],[0,0]])

cand = [[0,N]]

ans = 0
visited = set([])
while cand:
    #print(cand)
    temp = []
    for l,r in cand:
        if r-l < 2:
            continue
        if (l,r) in visited:
            continue
        visited.add((l,r))
        (m1,i1), (m2,i2) = seg.get(l,r)
        #print(P[l:r],m1,m2)
        i1,i2 = (i1,i2) if i1 < i2 else (i2,i1)
        ans += m2 * (i1-l+1) * (r-i2)
        temp.append([l,i2])
        temp.append([i1+1,r])
    cand = temp

print(ans)
Buggy python submission (verdict: Time Limit Exceeded):
from collections import Counter
N = int(input())
S = list(map(int,input().split()))
count = Counter(S)
S = sorted(S)
T = sorted(list(set(S)))
M = len(T)
pre = {T[i]:T[i-1] if i > 0 else 0 for i in range(M)}

l = [S[-1]]
count[S[-1]] -= 1

while len(l) < 2**N:
    for n in l[:]:
        m = pre[n]
        if m == 0:
            print('No')
            exit()
        while count[m] == 0:
            pre[n] = pre[m]
            m = pre[n]
            if m == 0:
                print('No')
                exit()
        l.append(m)
        count[m] -= 1

print('Yes')



Fixed submission:
from collections import Counter
N = int(input())
S = list(map(int,input().split()))
#N = 18
#S = list(range(1,2**N+1))

def solve():
    global S
    S += [0]
    count = Counter(S)
    S = sorted(S)
    T = sorted(list(set(S)))
    M = len(T)
    pre = {T[i]:T[i-1] if i > 0 else 0 for i in range(M)}

    l = [S[-1]]
    count[S[-1]] -= 1

    while len(l) < 2**N:
        for n in l[:]:
            m = pre[n]
            pl = []
            while count[m] == 0:
                pl.append(m)
                m = pre[m]
            if pl:
                pre[n] = m
                for p in pl:
                    pre[p] = m
            if m == 0:
                print('No')
                exit()
            l.append(m)
            count[m] -= 1

    print('Yes')
    #print(count)

solve()
Buggy python submission (verdict: Time Limit Exceeded):
from collections import defaultdict
from collections import deque
N = int(input())
# A_i,j = A[i-1][j-1]
A = [list(map(int,input().split(' '))) for i in [0]*N]

dic = defaultdict(int)
ns = [0] * N

ls = range(1,N+1)
ans = 0
while(True):
	new_ls = set([])
	for i in ls:
		if ns[i-1] >= N-1:
			continue
		j = A[i-1][ns[i-1]]
		dic[(i*j,i+j)] += 1
		if dic[(i*j,i+j)] == 2:
			new_ls.add(i)
			new_ls.add(j)
	if not new_ls:
		break
	for ind in new_ls:
		ns[ind-1] += 1
	ls = new_ls
	ans += 1

for i in range(N):
	if ns[i] != N-1:
		print(-1)
		exit()
print(ans)








Fixed submission:
from collections import defaultdict
from collections import deque
N = int(input())
# A_i,j = A[i-1][j-1]
A = [list(map(int,input().split(' '))) for i in [0]*N]

dic = defaultdict(int)
ns = [0] * N

ls = range(1,N+1)
ans = 0
while(True):
	new_ls = set([])
	for i in ls:
		if ns[i-1] >= N-1:
			continue
		j = A[i-1][ns[i-1]]
		dic[(i*j,i+j)] += 1
		if dic[(i*j,i+j)] == 2:
			new_ls.add(i)
			new_ls.add(j)
	if not new_ls:
		break
	for ind in new_ls:
		ns[ind-1] += 1
	ls = new_ls
	ans += 1

for i in range(N):
	if ns[i] != N-1:
		print(-1)
		exit()
print(ans)
Buggy python submission (verdict: Runtime Error):
N,Q = map(int,input().split(' '))

tree = {i:[] for i in range(1,N+1)}
sub = {i:[i] for i in range(1,N+1)}
fin = [False]*(N+1)
vals = [0]*(N+1)

for i in [0]*(N-1):
	a,b = map(int,input().split(' '))
	tree[a].append(b)

def get_sub(p):
	if not fin[p]:
		for t in tree[p]:
			sub[p].extend(get_sub(t))
		fin[p] = True
	return sub[p]

for i in [0]*Q:
	p,x = map(int,input().split(' '))
	subtree = get_sub(p)
	for t in subtree:
		vals[t] += x
#print(tree)
#print(sub)

print(' '.join(map(str,vals[1:])))






Fixed submission:
from collections import deque
N,Q = map(int,input().split(' '))

tree = {i:[] for i in range(1,N+1)}
#sub = {i:[i] for i in range(1,N+1)}
#fin = [False]*(N+1)
vals = [0]*(N+1)
#parent = [0]*(N+1)

for i in [0]*(N-1):
	a,b = map(int,input().split(' '))
	tree[a].append(b)
	#parent[b] = a

for i in [0]*Q:
	p,x = map(int,input().split(' '))
	vals[p] += x

ans = [0]*(N+1)

ans[1] = vals[1]

dq = deque([1])
while(dq):
	n = dq.popleft()
	for t in tree[n]:
		ans[t] = vals[t]+ans[n]
		dq.append(t)

print(' '.join(map(str,ans[1:])))
Buggy python submission (verdict: Time Limit Exceeded):
import bisect
N = int(input())
A = [int(input()) for i in [0]*N]

ls = [A[0]]
num = 1
for a in A[1:]:
	new = True 
	for i in range(num):
		if ls[i] < a:
			new = False
			ls[i] = a
			break
	if new:
		num += 1
		ls.append(a)

print(len(ls))






Fixed submission:
import bisect
from collections import deque
N = int(input())
A = [int(input()) for i in [0]*N]


ls = deque([A[0]])
for a in A[1:]:
	ind = bisect.bisect_left(ls,a)
	if ind == 0:
		ls.appendleft(a)
	else:
		ls[ind-1] = a

print(len(ls))
Buggy python submission (verdict: Runtime Error):
MOD = 10**9 + 7
N,M= map(int,input().split())
S= list(map(int,input().split()))
T= list(map(int,input().split()))

def dpinit(ps, val=0):
    res = [val for i in [0]*ps[-1]]
    for i in ps[:-1][::-1]:
        res = [res.copy() for k in [0]*i]
    return res

dp = dpinit((N+1,M+1))
for i in range(N+1):
    dp[i][0] = 1
for j in range(M+1):
    dp[0][j] = 1

for i in range(1,N+1):
    s = S[i-1]
    for j in range(1,M+1):
        t = T[j-1]
        dp[i][j] += dp[i-1][j] + dp[i][j-1]
        if s != t:
            dp[i][j] -= dp[i-1][j-1]
        dp[i][j] %= MOD

print(dp[N][M])




Fixed submission:
MOD = 10**9 + 7
N,M= map(int,input().split())
S= list(map(int,input().split()))
T= list(map(int,input().split()))

def dpinit(ps, val=0):
    res = [val for i in [0]*ps[-1]]
    for i in ps[:-1][::-1]:
        res = [res[:] for k in [0]*i]
    return res

dp = dpinit((N+1,M+1))
for i in range(N+1):
    dp[i][0] = 1
for j in range(M+1):
    dp[0][j] = 1

for i in range(1,N+1):
    s = S[i-1]
    for j in range(1,M+1):
        t = T[j-1]
        dp[i][j] += dp[i-1][j] + dp[i][j-1]
        if s != t:
            dp[i][j] -= dp[i-1][j-1]
        dp[i][j] %= MOD

print(dp[N][M])