#pragma GCC optimize("O2") #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include //#define int ll #define INT128_MAX (__int128)(((unsigned __int128) 1 << ((sizeof(__int128) * __CHAR_BIT__) - 1)) - 1) #define INT128_MIN (-INT128_MAX - 1) #ifdef DEBUG #define dbg(x) cout << (#x) << " = " << x << '\n' #else #define dbg(x) #endif namespace R = std::ranges; namespace V = std::views; using namespace std; using ll = long long; using ull = unsigned long long; using ldb = long double; //#define double ldb template ostream& operator<<(ostream& os, const array &arr) { for(const T &X : arr) os << X << ' '; return os; } template ostream& operator<<(ostream& os, const vector &vec) { for(const T &X : vec) os << X << ' '; return os; } template ostream& operator<<(ostream& os, const set &s) { for(const T &x : s) os << x << ' '; return os; } /** * template name: MontgomeryModInt * author: Misuki * reference: https://github.com/NyaanNyaan/library/blob/master/modint/montgomery-modint.hpp#L10 * last update: 2023/11/30 * note: mod should be a prime less than 2^30. */ template struct MontgomeryModInt { using mint = MontgomeryModInt; using i32 = int32_t; using u32 = uint32_t; using u64 = uint64_t; static constexpr u32 get_r() { u32 res = 1, base = mod; for(i32 i = 0; i < 31; i++) res *= base, base *= base; return -res; } static constexpr u32 get_mod() { return mod; } static constexpr u32 n2 = -u64(mod) % mod; //2^64 % mod static constexpr u32 r = get_r(); //-P^{-1} % 2^32 u32 a; static u32 reduce(const u64 &b) { return (b + u64(u32(b) * r) * mod) >> 32; } static u32 transform(const u64 &b) { return reduce(u64(b) * n2); } MontgomeryModInt() : a(0) {} MontgomeryModInt(const int64_t &b) : a(transform(b % mod + mod)) {} mint pow(u64 k) const { mint res(1), base(*this); while(k) { if (k & 1) res *= base; base *= base, k >>= 1; } return res; } mint inverse() const { return (*this).pow(mod - 2); } u32 get() const { u32 res = reduce(a); return res >= mod ? res - mod : res; } mint& operator+=(const mint &b) { if (i32(a += b.a - 2 * mod) < 0) a += 2 * mod; return *this; } mint& operator-=(const mint &b) { if (i32(a -= b.a) < 0) a += 2 * mod; return *this; } mint& operator*=(const mint &b) { a = reduce(u64(a) * b.a); return *this; } mint& operator/=(const mint &b) { a = reduce(u64(a) * b.inverse().a); return *this; } mint operator-() { return mint() - mint(*this); } bool operator==(mint b) const { return (a >= mod ? a - mod : a) == (b.a >= mod ? b.a - mod : b.a); } bool operator!=(mint b) const { return (a >= mod ? a - mod : a) != (b.a >= mod ? b.a - mod : b.a); } friend mint operator+(mint a, mint b) { return a += b; } friend mint operator-(mint a, mint b) { return a -= b; } friend mint operator*(mint a, mint b) { return a *= b; } friend mint operator/(mint a, mint b) { return a /= b; } friend ostream& operator<<(ostream& os, const mint& b) { return os << b.get(); } friend istream& operator>>(istream& is, mint& b) { int64_t val; is >> val; b = mint(val); return is; } }; using mint = MontgomeryModInt<998244353>; /** * template name: comb * author: Misuki * last update: 2023/01/22 * note: remember to call init() before using it. */ const int MAX = 200002; mint fac[MAX], facInv[MAX], pow2[MAX]; void init() { fac[0] = 1; for(int i = 1; i < MAX; i++) fac[i] = fac[i - 1] * i; facInv[MAX - 1] = 1 / fac[MAX - 1]; for(int i = MAX - 2; i >= 0; i--) facInv[i] = facInv[i + 1] * (i + 1); pow2[0] = 1; for(int i = 1; i < MAX; i++) pow2[i] = pow2[i - 1] * 2; } mint binom(int a, int b) { if (b < 0 or a < b) return 0; else return fac[a] * facInv[b] * facInv[a - b]; } mint C(int k) { return fac[k] * facInv[k + 1] * binom(2 * k, k); } /** * template name: NTTmint * reference: https://judge.yosupo.jp/submission/69896 * last update: 2024/01/07 * include: mint * remark: MOD = 2^K * C + 1, R is a primitive root modulo MOD * remark: a.size() <= 2^K must be satisfied * some common modulo: 998244353 = 2^23 * 119 + 1, R = 3 * 469762049 = 2^26 * 7 + 1, R = 3 * 1224736769 = 2^24 * 73 + 1, R = 3 * verify: Library Checker - Convolution */ template> struct NTT { using u32 = uint32_t; static constexpr u32 mod = (1 << k) * c + 1; static constexpr u32 get_mod() { return mod; } static void ntt(vector &a, bool inverse) { static array w, w_inv; if (w[0] == 0) { Mint root = 2; while(root.pow((mod - 1) / 2) == 1) root += 1; for(int i = 0; i < 30; i++) w[i] = -(root.pow((mod - 1) >> (i + 2))), w_inv[i] = 1 / w[i]; } int n = ssize(a); if (not inverse) { for(int m = n; m >>= 1; ) { Mint ww = 1; for(int s = 0, l = 0; s < n; s += 2 * m) { for(int i = s, j = s + m; i < s + m; i++, j++) { Mint x = a[i], y = a[j] * ww; a[i] = x + y, a[j] = x - y; } ww *= w[__builtin_ctz(++l)]; } } } else { for(int m = 1; m < n; m *= 2) { Mint ww = 1; for(int s = 0, l = 0; s < n; s += 2 * m) { for(int i = s, j = s + m; i < s + m; i++, j++) { Mint x = a[i], y = a[j]; a[i] = x + y, a[j] = (x - y) * ww; } ww *= w_inv[__builtin_ctz(++l)]; } } Mint inv = 1 / Mint(n); for(Mint &x : a) x *= inv; } } static vector conv(vector a, vector b) { int sz = ssize(a) + ssize(b) - 1; int n = bit_ceil((u32)sz); a.resize(n, 0); ntt(a, false); b.resize(n, 0); ntt(b, false); for(int i = 0; i < n; i++) a[i] *= b[i]; ntt(a, true); a.resize(sz); return a; } }; /** * template name: FPS * author: Misuki * last update: 2024/01/10 * include: NTT/mint * verify: Library Checker - Inv of Formal Power Series * Library Checker - Exp of Formal Power Series * Library Checker - Log of Formal Power Series * Library Checker - Pow of Formal Power Series * Library Checker - Convolution * Library Checker - Division of Polynomials * Library Checker - Multipoint Evaluation * Library Checker - Polynomial Interpolation */ template struct FPS : vector { static function(vector, vector)> conv; FPS(vector v) { *this = v; } using vector::vector; FPS& operator+=(FPS b) { if (ssize(*this) < ssize(b)) this -> resize(ssize(b), 0); for(int i = 0; i < ssize(b); i++) (*this)[i] += b[i]; return *this; } FPS& operator-=(FPS b) { if (ssize(*this) < ssize(b)) this -> resize(ssize(b), 0); for(int i = 0; i < ssize(b); i++) (*this)[i] -= b[i]; return *this; } FPS& operator*=(FPS b) { auto c = conv(*this, b); this -> resize(ssize(*this) + ssize(b) - 1); copy(c.begin(), c.end(), this -> begin()); return *this; } FPS& operator*=(Mint b) { for(int i = 0; i < ssize(*this); i++) (*this)[i] *= b; return *this; } FPS& operator/=(Mint b) { b = Mint(1) / b; for(int i = 0; i < ssize(*this); i++) (*this)[i] *= b; return *this; } FPS shrink() { FPS F = *this; while(ssize(F) > 1 and F.back() == 0) F.pop_back(); return F; } FPS integral() { vector Inv(ssize(*this) + 1); Inv[1] = 1; for(int i = 2; i < ssize(Inv); i++) Inv[i] = (Mint::get_mod() - Mint::get_mod() / i) * Inv[Mint::get_mod() % i]; FPS Q(ssize(*this) + 1, 0); for(int i = 0; i < ssize(*this); i++) Q[i + 1] = (*this)[i] * Inv[i + 1]; return Q; } FPS derivative() { assert(!this -> empty()); FPS Q(ssize(*this) - 1); for(int i = 1; i < ssize(*this); i++) Q[i - 1] = (*this)[i] * i; return Q; } Mint eval(Mint x) { Mint base = 1, res = 0; for(int i = 0; i < ssize(*this); i++, base *= x) res += (*this)[i] * base; return res; } FPS inv(int k) { // 1 / FPS (mod x^k) assert(!this -> empty() and (*this)[0] != 0); FPS Q(1, 1 / (*this)[0]); for(int i = 1; (1 << (i - 1)) < k; i++) { FPS P = (*this); P.resize(1 << i, 0); Q = Q * (FPS(1, 2) - P * Q); Q.resize(1 << i, 0); } Q.resize(k); return Q; } array div(FPS G) { FPS F = this -> shrink(); G = G.shrink(); assert(!G.empty()); if (ssize(G) > ssize(F)) return {{{}, F}}; int n = ssize(F) - ssize(G) + 1; auto FR = F, GR = G; R::reverse(FR); R::reverse(GR); FPS Q = FR * GR.inv(n); Q.resize(n); R::reverse(Q); return {Q, (F - G * Q).shrink()}; } FPS log(int k) { assert(!this -> empty() and (*this)[0] == 1); FPS Q = *this; Q = (Q.derivative() * Q.inv(k)); Q.resize(k - 1); return Q.integral(); } FPS exp(int k) { assert(!this -> empty() and (*this)[0] == 0); FPS Q(1, 1); for(int i = 1; (1 << (i - 1)) < k; i++) { FPS P = (*this); P.resize(1 << i, 0); Q = Q * (FPS(1, 1) + P - Q.log(1 << i)); Q.resize(1 << i, 0); } Q.resize(k); return Q; } FPS pow(ll idx, int k) { if (idx == 0) { FPS res(k, 0); res[0] = 1; return res; } for(int i = 0; i < ssize(*this) and i * idx < k; i++) { if ((*this)[i] != 0) { Mint Inv = 1 / (*this)[i]; FPS Q(ssize(*this) - i); for(int j = i; j < ssize(*this); j++) Q[j - i] = (*this)[j] * Inv; Q = (Q.log(k) * idx).exp(k); FPS Q2(k, 0); Mint Pow = (*this)[i].pow(idx); for(int j = 0; j + i * idx < k; j++) Q2[j + i * idx] = Q[j] * Pow; return Q2; } } return FPS(k, 0); } vector multieval(vector xs) { int n = ssize(xs); vector data(2 * n); for(int i = 0; i < n; i++) data[n + i] = {-xs[i], 1}; for(int i = n - 1; i > 0; i--) data[i] = data[i << 1] * data[i << 1 | 1]; data[1] = (this -> div(data[1]))[1]; for(int i = 1; i < n; i++) { data[i << 1] = data[i].div(data[i << 1])[1]; data[i << 1 | 1] = data[i].div(data[i << 1 | 1])[1]; } vector res(n); for(int i = 0; i < n; i++) res[i] = data[n + i][0]; return res; } static vector interpolate(vector xs, vector ys) { assert(ssize(xs) == ssize(ys)); int n = ssize(xs); vector data(2 * n), res(2 * n); for(int i = 0; i < n; i++) data[n + i] = {-xs[i], 1}; for(int i = n - 1; i > 0; i--) data[i] = data[i << 1] * data[i << 1 | 1]; res[1] = data[1].derivative().div(data[1])[1]; for(int i = 1; i < n; i++) { res[i << 1] = res[i].div(data[i << 1])[1]; res[i << 1 | 1] = res[i].div(data[i << 1 | 1])[1]; } for(int i = 0; i < n; i++) res[n + i][0] = ys[i] / res[n + i][0]; for(int i = n - 1; i > 0; i--) res[i] = res[i << 1] * data[i << 1 | 1] + res[i << 1 | 1] * data[i << 1]; return res[1]; } friend FPS operator+(FPS a, FPS b) { return a += b; } friend FPS operator-(FPS a, FPS b) { return a -= b; } friend FPS operator*(FPS a, FPS b) { return a *= b; } friend FPS operator*(FPS a, Mint b) { return a *= b; } friend FPS operator/(FPS a, Mint b) { return a /= b; } }; NTT ntt; using fps = FPS; template<> function(vector, vector)> fps::conv = ntt.conv; signed main() { ios::sync_with_stdio(false), cin.tie(NULL); init(); int n, m; cin >> n >> m; vector poly; while(m--) { int l, r; cin >> l >> r; if ((r - l + 1) % 2 == 0) { poly.push_back(fps(r - l + 2, 0)); poly.back()[0] = 1, poly.back().back() = -C((r - l + 1) / 2); } } auto calc = [&](int l, int r, auto self) -> fps { if (l + 1 == r) return poly[l]; else return self(l, (l + r) / 2, self) * self((l + r) / 2, r, self); }; auto f = calc(0, ssize(poly), calc); mint ans = 0; for(int i = 0; i < ssize(f); i += 2) ans += f[i] * C((n - i) / 2); cout << ans << '\n'; return 0; }