#include #include #define rep(i,n) for(int i=0;i vi; typedef vector vl; typedef vector> vvi; typedef vector> vvl; typedef long double ld; typedef pair P; template ostream& operator<<(ostream& os, const static_modint& a) {os << a.val(); return os;} template ostream& operator<<(ostream& os, const dynamic_modint& a) {os << a.val(); return os;} template istream& operator>>(istream& is, static_modint& a) {long long x; is >> x; a = x; return is;} template istream& operator>>(istream& is, dynamic_modint& a) {long long x; is >> x; a = x; return is;} template istream& operator>>(istream& is, vector& v){int n = v.size(); assert(n > 0); rep(i, n) is >> v[i]; return is;} template ostream& operator<<(ostream& os, const pair& p){os << p.first << ' ' << p.second; return os;} template ostream& operator<<(ostream& os, const vector& v){int n = v.size(); rep(i, n) os << v[i] << (i == n - 1 ? "\n" : " "); return os;} template ostream& operator<<(ostream& os, const vector>& v){int n = v.size(); rep(i, n) os << v[i] << (i == n - 1 ? "\n" : ""); return os;} template ostream& operator<<(ostream& os, const set& se){for(T x : se) os << x << " "; os << "\n"; return os;} template ostream& operator<<(ostream& os, const unordered_set& se){for(T x : se) os << x << " "; os << "\n"; return os;} template ostream& operator<<(ostream& os, const atcoder::segtree& seg){int n = seg.max_right(0, [](S){return true;}); rep(i, n) os << seg.get(i) << (i == n - 1 ? "\n" : " "); return os;} template ostream& operator<<(ostream& os, const atcoder::lazy_segtree& seg){int n = seg.max_right(0, [](S){return true;}); rep(i, n) os << seg.get(i) << (i == n - 1 ? "\n" : " "); return os;} template void chmin(T& a, T b){a = min(a, b);} template void chmax(T& a, T b){a = max(a, b);} random_device rd; mt19937 mt(rd()); int random_prime(){ while (true){ int pr = mt() % 100000000 + 900000000; if (atcoder::internal::is_prime_constexpr(pr)){ return pr; } } } using ull = unsigned long long; using mint1 = atcoder::dynamic_modint<1>; using mint2 = atcoder::dynamic_modint<2>; struct mints{ mint1 d1; mint2 d2; mints(long long val = 0) : d1(val), d2(val) {} mints(mint1 d1, mint2 d2) : d1(d1), d2(d2) {} mints& operator++() { d1++; d2++; return *this; } mints& operator--() { d1--; d2--; return *this; } mints& operator+=(const mints& a) { d1 += a.d1; d2 += a.d2; return *this; } mints& operator-=(const mints& a) { d1 -= a.d1; d2 -= a.d2; return *this; } mints& operator*=(const mints& a) { d1 *= a.d1; d2 *= a.d2; return *this; } mints& operator/=(const mints& a) { return *this = *this * a.inv(); } mints operator+() const { return *this; } mints operator-() const { return mints(0) - *this; } mints pow(long long n) const { return mints(d1.pow(n), d2.pow(n)); } mints inv() const { return mints(d1.inv(), d2.inv()); } mints operator+(const mints& a) const { return mints(d1 + a.d1, d2 + a.d2); } mints operator-(const mints& a) const { return mints(d1 - a.d1, d2 - a.d2); } mints operator*(const mints& a) const { return mints(d1 * a.d1, d2 * a.d2); } mints operator/(const mints& a) const { return mints(d1 / a.d1, d2 / a.d2); } bool operator==(const mints& a) const { return d1 == a.d1 and d2 == a.d2; } bool operator!=(const mints& a) const { return d1 != a.d1 or d2 != a.d2; } ull val(){ return (ull(d1.val()) << 32) + ull(d2.val()); }; }; ostream& operator<<(ostream& os, const mints& a) {os << a.d1 << '&' << a.d2; return os;} void random_mod(){ const int p1 = random_prime(); const int p2 = random_prime(); mint1::set_mod(p1); mint2::set_mod(p2); } long long MOD; int K; vector S; vector T; void input(){ cin >> K; S.resize(K); T.resize(K); rep(i, K) cin >> S[i] >> T[i]; } template T primitive(){ T g = 0; int p = T::mod(); auto factor = [&](int x){ assert(x >= 1); vector res; for(int y = 1; y * y <= x; y++){ if(x % y == 0){ res.push_back(y); res.push_back(x / y); } } return res; }; vector factor_list = factor(p- 1); while(true){ g = mt() % (p - 1) + 1; bool success = true; for(int y : factor_list){ if(y == p - 1) continue; T h = g.pow(y); if(h == T(1)) success = false; } if(success) break; } return g; } bool judge_palindrome(){ using F = mints; vector hash(2, 0); F base(primitive(), primitive()); auto get_hash = [&]() -> F{ F x(0); rep(i, K){ F p = F(base).pow(int(S[i].size())); F q = p.pow(T[i]); x *= q; F y(0); for(auto c : S[i]){ y *= base; y += c - '0' + 1; } x += y * (q - 1) / (p - 1); } return x; }; rep(t, 2){ hash[t] = get_hash(); reverse(S.begin(), S.end()); reverse(T.begin(), T.end()); rep(i, K) reverse(S[i].begin(), S[i].end()); } return hash[0] == hash[1]; } int main() { random_mod(); input(); bool is_palindrome = judge_palindrome(); if(is_palindrome) cout << "Yes\n"; else cout << "No\n"; return 0; }