from sys import exit, stdin, setrecursionlimit from collections import deque, defaultdict, Counter from copy import deepcopy from bisect import bisect_left, bisect_right, insort_left, insort_right from heapq import heapify, heappop, heappush from itertools import product, permutations, combinations, combinations_with_replacement from functools import reduce from math import gcd, sin, cos, tan, asin, acos, atan, atan2, degrees, radians, ceil, floor, sqrt, factorial from math import pi as PI from random import randint # from decimal import Decimal setrecursionlimit(500000) INF = (1<<61)-1 EPS = 1e-10 MOD = 10**9+7 # MOD = 998244353 def input(): return stdin.readline()[:-1] def intput(): return int(stdin.readline()) def minput(): return stdin.readline()[:-1].split() def linput(): return stdin.readline()[:-1].split() def mint(): return map(int,stdin.readline().split()) def lint(): return list(map(int,stdin.readline().split())) def ilint(): return int(stdin.readline()), list(map(int,stdin.readline().split())) def lcm(x,y): return x*y//gcd(x,y) def lgcd(l): return reduce(gcd,l) def llcm(l): return reduce(lcm,l) def powmod(n,i,mod=MOD): return pow(n,mod-1+i,mod) if i<0 else pow(n,i,mod) def div2(x): return x.bit_length() def div10(x): return len(str(x))-(x==0) def popcount(x): return bin(x).count('1') def digit(x,i,max_len=None): s = str(x) if max_len: i -= max_len-len(s) return int(s[i-1]) if i>0 else 0 def digitsum(x): ans = 0 for i in range(div10(x)): ans += digit(x,i+1) return ans def pf(x,mode='counter'): C = Counter() p = 2 while x>1: k = 0 while x%p==0: x //= p k += 1 if k>0: C[p] += k p = p+2-(p==2) if p*p> 15, A & (1 << 15) - 1 B1, B2 = B >> 15, B & (1 << 15) - 1 X = np.dot(A1, B1) % MOD Y = np.dot(A2, B2) Z = np.dot(A1 + A2, B1 + B2) - X - Y return ((X << 30) + (Z << 15) + Y) % MOD def matpow(A, N): P = np.eye(A.shape[0], dtype=np.int64) while N: if N & 1: P = matmul(P, A) A = matmul(A, A) N >>= 1 return P def zash(S): lis = sorted(S) dic = {} for i,x in enumerate(lis): dic[x] = i return lis, dic def pr(*x): print(*x, sep='', end='') if len(x) else print() def lprint(l): for x in l: print(x) def ston(c, c0='a'): return ord(c)-ord(c0) def ntos(x, c0='a'): return chr(x+ord(c0)) def judge(x, l=['Yes', 'No']): print(l[0] if x else l[1]) def debug(*x, flag=1): if flag: print(*x) ###################################################### SIZE=4*10**5 comb_table = [1]*(SIZE+1) inv_table = [1]*(SIZE+1) combinv_table = [1]*(SIZE+1) for i in range(2,SIZE+1): comb_table[i] = (comb_table[i-1] * i) % MOD inv_table[i] = (-inv_table[MOD%i] * (MOD//i)) % MOD combinv_table[i] = (combinv_table[i-1] * inv_table[i]) % MOD def comb(n,k): if k<0 or n