#line 1 "/home/maspy/compro/library/my_template.hpp" #if defined(LOCAL) #include #else // https://codeforces.com/blog/entry/96344 #pragma GCC optimize("Ofast,unroll-loops") #pragma GCC target("avx2,popcnt") #include using namespace std; using ll = long long; using u32 = unsigned int; using u64 = unsigned long long; template constexpr T infty = 0; template <> constexpr int infty = 1'000'000'000; template <> constexpr ll infty = ll(infty) * infty * 2; template <> constexpr u32 infty = infty; template <> constexpr u64 infty = infty; template <> constexpr double infty = infty; template <> constexpr long double infty = infty; using pi = pair; using vi = vector; template using vc = vector; template using vvc = vector>; template using vvvc = vector>; template using vvvvc = vector>; template using vvvvvc = vector>; template using pq = priority_queue; template using pqg = priority_queue, greater>; #define vv(type, name, h, ...) \ vector> name(h, vector(__VA_ARGS__)) #define vvv(type, name, h, w, ...) \ vector>> name( \ h, vector>(w, vector(__VA_ARGS__))) #define vvvv(type, name, a, b, c, ...) \ vector>>> name( \ a, vector>>( \ b, vector>(c, vector(__VA_ARGS__)))) // https://trap.jp/post/1224/ #define FOR1(a) for (ll _ = 0; _ < ll(a); ++_) #define FOR2(i, a) for (ll i = 0; i < ll(a); ++i) #define FOR3(i, a, b) for (ll i = a; i < ll(b); ++i) #define FOR4(i, a, b, c) for (ll i = a; i < ll(b); i += (c)) #define FOR1_R(a) for (ll i = (a)-1; i >= ll(0); --i) #define FOR2_R(i, a) for (ll i = (a)-1; i >= ll(0); --i) #define FOR3_R(i, a, b) for (ll i = (b)-1; i >= ll(a); --i) #define overload4(a, b, c, d, e, ...) e #define overload3(a, b, c, d, ...) d #define FOR(...) overload4(__VA_ARGS__, FOR4, FOR3, FOR2, FOR1)(__VA_ARGS__) #define FOR_R(...) overload3(__VA_ARGS__, FOR3_R, FOR2_R, FOR1_R)(__VA_ARGS__) #define FOR_subset(t, s) \ for (ll t = (s); t >= 0; t = (t == 0 ? -1 : (t - 1) & (s))) #define all(x) x.begin(), x.end() #define len(x) ll(x.size()) #define elif else if #define eb emplace_back #define mp make_pair #define mt make_tuple #define fi first #define se second #define stoi stoll int popcnt(int x) { return __builtin_popcount(x); } int popcnt(u32 x) { return __builtin_popcount(x); } int popcnt(ll x) { return __builtin_popcountll(x); } int popcnt(u64 x) { return __builtin_popcountll(x); } int popcnt_mod_2(int x) { return __builtin_parity(x); } int popcnt_mod_2(u32 x) { return __builtin_parity(x); } int popcnt_mod_2(ll x) { return __builtin_parityll(x); } int popcnt_mod_2(u64 x) { return __builtin_parityll(x); } // (0, 1, 2, 3, 4) -> (-1, 0, 1, 1, 2) int topbit(int x) { return (x == 0 ? -1 : 31 - __builtin_clz(x)); } int topbit(u32 x) { return (x == 0 ? -1 : 31 - __builtin_clz(x)); } int topbit(ll x) { return (x == 0 ? -1 : 63 - __builtin_clzll(x)); } int topbit(u64 x) { return (x == 0 ? -1 : 63 - __builtin_clzll(x)); } // (0, 1, 2, 3, 4) -> (-1, 0, 1, 0, 2) int lowbit(int x) { return (x == 0 ? -1 : __builtin_ctz(x)); } int lowbit(u32 x) { return (x == 0 ? -1 : __builtin_ctz(x)); } int lowbit(ll x) { return (x == 0 ? -1 : __builtin_ctzll(x)); } int lowbit(u64 x) { return (x == 0 ? -1 : __builtin_ctzll(x)); } template T floor(T a, T b) { return a / b - (a % b && (a ^ b) < 0); } template T ceil(T x, T y) { return floor(x + y - 1, y); } template T bmod(T x, T y) { return x - y * floor(x, y); } template pair divmod(T x, T y) { T q = floor(x, y); return {q, x - q * y}; } template T SUM(const vector &A) { T sm = 0; for (auto &&a: A) sm += a; return sm; } #define MIN(v) *min_element(all(v)) #define MAX(v) *max_element(all(v)) #define LB(c, x) distance((c).begin(), lower_bound(all(c), (x))) #define UB(c, x) distance((c).begin(), upper_bound(all(c), (x))) #define UNIQUE(x) \ sort(all(x)), x.erase(unique(all(x)), x.end()), x.shrink_to_fit() template T POP(deque &que) { T a = que.front(); que.pop_front(); return a; } template T POP(pq &que) { T a = que.top(); que.pop(); return a; } template T POP(pqg &que) { T a = que.top(); que.pop(); return a; } template T POP(vc &que) { T a = que.back(); que.pop_back(); return a; } template ll binary_search(F check, ll ok, ll ng, bool check_ok = true) { if (check_ok) assert(check(ok)); while (abs(ok - ng) > 1) { auto x = (ng + ok) / 2; (check(x) ? ok : ng) = x; } return ok; } template double binary_search_real(F check, double ok, double ng, int iter = 100) { FOR(iter) { double x = (ok + ng) / 2; (check(x) ? ok : ng) = x; } return (ok + ng) / 2; } template inline bool chmax(T &a, const S &b) { return (a < b ? a = b, 1 : 0); } template inline bool chmin(T &a, const S &b) { return (a > b ? a = b, 1 : 0); } // ? は -1 vc s_to_vi(const string &S, char first_char) { vc A(S.size()); FOR(i, S.size()) { A[i] = (S[i] != '?' ? S[i] - first_char : -1); } return A; } template vector cumsum(vector &A, int off = 1) { int N = A.size(); vector B(N + 1); FOR(i, N) { B[i + 1] = B[i] + A[i]; } if (off == 0) B.erase(B.begin()); return B; } // stable sort template vector argsort(const vector &A) { vector ids(len(A)); iota(all(ids), 0); sort(all(ids), [&](int i, int j) { return (A[i] == A[j] ? i < j : A[i] < A[j]); }); return ids; } // A[I[0]], A[I[1]], ... template vc rearrange(const vc &A, const vc &I) { vc B(len(I)); FOR(i, len(I)) B[i] = A[I[i]]; return B; } #endif #line 1 "/home/maspy/compro/library/other/io2.hpp" #define INT(...) \ int __VA_ARGS__; \ IN(__VA_ARGS__) #define LL(...) \ ll __VA_ARGS__; \ IN(__VA_ARGS__) #define STR(...) \ string __VA_ARGS__; \ IN(__VA_ARGS__) #define CHR(...) \ char __VA_ARGS__; \ IN(__VA_ARGS__) #define DBL(...) \ long double __VA_ARGS__; \ IN(__VA_ARGS__) #define VEC(type, name, size) \ vector name(size); \ read(name) #define VV(type, name, h, w) \ vector> name(h, vector(w)); \ read(name) void read(int &a) { cin >> a; } void read(long long &a) { cin >> a; } void read(char &a) { cin >> a; } void read(double &a) { cin >> a; } void read(long double &a) { cin >> a; } void read(string &a) { cin >> a; } template void read(pair &p) { read(p.first), read(p.second); } template void read(vector &a) {for(auto &i : a) read(i);} template void read(T &a) { cin >> a; } void IN() {} template void IN(Head &head, Tail &...tail) { read(head); IN(tail...); } template ostream& operator<<(ostream& os, const pair& A) { os << A.fi << " " << A.se; return os; } template ostream& operator<<(ostream& os, const vector& A) { for (size_t i = 0; i < A.size(); i++) { if(i) os << " "; os << A[i]; } return os; } void print() { cout << "\n"; cout.flush(); } template void print(Head&& head, Tail&&... tail) { cout << head; if (sizeof...(Tail)) cout << " "; print(forward(tail)...); } void YES(bool t = 1) { print(t ? "YES" : "NO"); } void NO(bool t = 1) { YES(!t); } void Yes(bool t = 1) { print(t ? "Yes" : "No"); } void No(bool t = 1) { Yes(!t); } void yes(bool t = 1) { print(t ? "yes" : "no"); } void no(bool t = 1) { yes(!t); } #line 2 "/home/maspy/compro/library/linalg/xor/transpose.hpp" // n x m 行列の transpose。O((n+m)log(n+m)) 時間。 // https://github.com/dsnet/matrix-transpose template vc transpose(int n, int m, vc& A, bool keep_A = 1) { assert(max(n, m) <= numeric_limits::digits); assert(len(A) == n); vc tmp; if (keep_A) tmp = A; int LOG = 0; while ((1 << LOG) < max(n, m)) ++LOG; A.resize(1 << LOG); int width = 1 << LOG; UINT mask = 1; FOR(i, LOG) mask = mask | (mask << (1 << i)); FOR(t, LOG) { width >>= 1; mask = mask ^ (mask >> width); FOR(i, 1 << t) { FOR(j, width) { UINT* x = &A[width * (2 * i + 0) + j]; UINT* y = &A[width * (2 * i + 1) + j]; *x = ((*y << width) & mask) ^ *x; *y = ((*x & mask) >> width) ^ *y; *x = ((*y << width) & mask) ^ *x; } } } A.resize(m); if (!keep_A) return A; swap(A, tmp); return tmp; } #line 2 "/home/maspy/compro/library/linalg/xor/vector_space.hpp" template struct Vector_Space { #define SP Vector_Space vc dat; Vector_Space() {} Vector_Space(vc dat, bool is_reduced = false) : dat(dat) { if (!is_reduced) reduce(); } int size() { return dat.size(); } int dim() { return dat.size(); } bool add_element(UINT v) { for (auto&& e: dat) { if (e == 0 || v == 0) break; chmin(v, v ^ e); } if (v) { dat.eb(v); return true; } return false; } bool contain(UINT v) { for (auto&& w: dat) { if (v == 0) break; chmin(v, v ^ w); } return v == 0; } UINT get_max(UINT xor_val = 0) { UINT res = xor_val; for (auto&& x: dat) chmax(res, res ^ x); return res; } UINT get_min(UINT xor_val) { UINT res = xor_val; for (auto&& x: dat) chmin(res, res ^ x); return res; } static SP merge(SP x, SP y) { if (len(x) < len(y)) swap(x, y); for (auto v: y.dat) { x.add_element(v); } return x; } static SP intersection(SP& x, SP& y, int max_dim) { SP xx = x.orthogonal_space(max_dim); SP yy = y.orthogonal_space(max_dim); xx = merge(xx, yy); return xx.orthogonal_space(max_dim); } SP orthogonal_space(int max_dim) { normalize(); int m = max_dim; // pivot[k] == k となるように行の順番を変える vc tmp(m); FOR(i, len(dat)) tmp[topbit(dat[i])] = dat[i]; tmp = transpose(m, m, tmp, 0); SP res; FOR(j, m) { if (tmp[j] >> j & 1) continue; res.add_element(tmp[j] | UINT(1) << j); } return res; } void normalize(bool dec = true) { int n = len(dat); // 三角化 FOR(j, n) FOR(i, j) chmin(dat[i], dat[i] ^ dat[j]); sort(all(dat)); if (dec) reverse(all(dat)); } private: void reduce() { SP y; for (auto&& e: dat) y.add_element(e); (*this) = y; } #undef SP }; #line 4 "main.cpp" void solve() { LL(N); VEC(int, A, N); ll s = 0; for (auto& x: A) s ^= x; if (s != 0) return No(); Vector_Space X; for (auto& x: A) X.add_element(x); ll dim = X.dim(); Yes(dim < N - 1); } signed main() { int T = 1; // INT(T); FOR(T) solve(); return 0; }