import io import sys from collections import defaultdict, deque, Counter from itertools import permutations, combinations, accumulate from heapq import heappush, heappop sys.setrecursionlimit(10**6) from bisect import bisect_right, bisect_left from math import gcd import math import atcoder _INPUT = """\ 6 5 3 2 6 7 19 14 1 4 2 3 5 4 1 5 7 5 4 7 6 5 4 3 1 5 2 2 3 3 2 4 4 1 3 5 3 1 1000 1000 100 1 3 10 """ def gcd(a, b): while b: a, b = b, a % b return a def lcm(a, b): return a // gcd(a, b) * b def isPrimeMR(n): d = n - 1 d = d // (d & -d) L = [2, 7, 61] if n < 1<<32 else [2, 3, 5, 7, 11, 13, 17] if n < 1<<48 else [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37] for a in L: t = d y = pow(a, t, n) if y == 1: continue while y != n - 1: y = y * y % n if y == 1 or t == n - 1: return 0 t <<= 1 return 1 def findFactorRho(n): m = 1 << n.bit_length() // 8 for c in range(1, 99): f = lambda x: (x * x + c) % n y, r, q, g = 2, 1, 1, 1 while g == 1: x = y for i in range(r): y = f(y) k = 0 while k < r and g == 1: ys = y for i in range(min(m, r - k)): y = f(y) q = q * abs(x - y) % n g = gcd(q, n) k += m r <<= 1 if g == n: g = 1 while g == 1: ys = f(ys) g = gcd(abs(x - ys), n) if g < n: if isPrimeMR(g): return g elif isPrimeMR(n // g): return n // g return findFactorRho(g) def primeFactor(n): i = 2 ret = {} rhoFlg = 0 while i * i <= n: k = 0 while n % i == 0: n //= i k += 1 if k: ret[i] = k i += i % 2 + (3 if i % 3 == 1 else 1) if i == 101 and n >= 2 ** 20: while n > 1: if isPrimeMR(n): ret[n], n = 1, 1 else: rhoFlg = 1 j = findFactorRho(n) k = 0 while n % j == 0: n //= j k += 1 ret[j] = k if n > 1: ret[n] = 1 if rhoFlg: ret = {x: ret[x] for x in sorted(ret)} return ret def divisors(N): pf = primeFactor(N) ret = [1] for p in pf: ret_prev = ret ret = [] for i in range(pf[p]+1): for r in ret_prev: ret.append(r * (p ** i)) return sorted(ret) import math def Eratosthenes(n): prime=[] furui=list(range(2,n+1)) while furui[0]