#include <bits/stdc++.h>
using i32 = int;
using u32 = unsigned int;
using i64 = long long;
using u64 = unsigned long long;
using i128 = __int128_t;
using u128 = __uint128_t;
using f64 = double;
using f80 = long double;
using f128 = __float128;
constexpr i32 operator"" _i32(u64 v) { return v; }
constexpr u32 operator"" _u32(u64 v) { return v; }
constexpr i64 operator"" _i64(u64 v) { return v; }
constexpr u64 operator"" _u64(u64 v) { return v; }
constexpr f64 operator"" _f64(f80 v) { return v; }
constexpr f80 operator"" _f80(f80 v) { return v; }
using Istream = std::istream;
using Ostream = std::ostream;
using Str = std::string;
template<typename T>
using Lt = std::less<T>;
template<typename T>
using Gt = std::greater<T>;
template<int n>
using BSet = std::bitset<n>;
template<typename T1, typename T2>
using Pair = std::pair<T1, T2>;
template<typename... Ts>
using Tup = std::tuple<Ts...>;
template<typename T, int N>
using Arr = std::array<T, N>;
template<typename... Ts>
using Deq = std::deque<Ts...>;
template<typename... Ts>
using Set = std::set<Ts...>;
template<typename... Ts>
using MSet = std::multiset<Ts...>;
template<typename... Ts>
using USet = std::unordered_set<Ts...>;
template<typename... Ts>
using UMSet = std::unordered_multiset<Ts...>;
template<typename... Ts>
using Map = std::map<Ts...>;
template<typename... Ts>
using MMap = std::multimap<Ts...>;
template<typename... Ts>
using UMap = std::unordered_map<Ts...>;
template<typename... Ts>
using UMMap = std::unordered_multimap<Ts...>;
template<typename... Ts>
using Vec = std::vector<Ts...>;
template<typename... Ts>
using Stack = std::stack<Ts...>;
template<typename... Ts>
using Queue = std::queue<Ts...>;
template<typename T>
using MaxHeap = std::priority_queue<T>;
template<typename T>
using MinHeap = std::priority_queue<T, Vec<T>, Gt<T>>;
constexpr bool LOCAL = false;
constexpr bool OJ = not LOCAL;
template<typename T>
static constexpr T OjLocal(T oj, T local)
{
    return LOCAL ? local : oj;
}
template<typename T>
constexpr T LIMMIN = std::numeric_limits<T>::min();
template<typename T>
constexpr T LIMMAX = std::numeric_limits<T>::max();
template<typename T = i64>
constexpr T INF = (LIMMAX<T> - 1) / 2;
template<typename T = f80>
constexpr T PI = T{3.141592653589793238462643383279502884};
template<typename T = u64>
constexpr T TEN(int n)
{
    return n == 0 ? T{1} : TEN<T>(n - 1) * T{10};
}
template<typename T>
constexpr bool chmin(T& a, const T& b)
{
    return (a > b ? (a = b, true) : false);
}
template<typename T>
constexpr bool chmax(T& a, const T& b)
{
    return (a < b ? (a = b, true) : false);
}
template<typename T>
constexpr T floorDiv(T x, T y)
{
    assert(y != 0);
    if (y < 0) { x = -x, y = -y; }
    return x >= 0 ? x / y : (x - y + 1) / y;
}
template<typename T>
constexpr T ceilDiv(T x, T y)
{
    assert(y != 0);
    if (y < 0) { x = -x, y = -y; }
    return x >= 0 ? (x + y - 1) / y : x / y;
}
template<typename T, typename I>
constexpr T powerMonoid(T v, I n, const T& e)
{
    assert(n >= 0);
    if (n == 0) { return e; }
    return (n % 2 == 1 ? v * powerMonoid(v, n - 1, e) : powerMonoid(v * v, n / 2, e));
}
template<typename T, typename I>
constexpr T powerInt(T v, I n)
{
    return powerMonoid(v, n, T{1});
}
template<typename Vs, typename... Args>
constexpr auto accumAll(const Vs& vs, Args... args)
{
    return std::accumulate(std::begin(vs), std::end(vs), args...);
}
template<typename Vs>
constexpr auto sumAll(const Vs& vs)
{
    return accumAll(vs, decltype(vs[0]){});
}
template<typename Vs>
constexpr auto gcdAll(const Vs& vs)
{
    return accumAll(vs, decltype(vs[0]){}, [&](auto v1, auto v2) { return std::gcd(v1, v2); });
}
template<typename Vs, typename V>
constexpr int lbInd(const Vs& vs, const V& v)
{
    return std::lower_bound(std::begin(vs), std::end(vs), v) - std::begin(vs);
}
template<typename Vs, typename V>
constexpr int ubInd(const Vs& vs, const V& v)
{
    return std::upper_bound(std::begin(vs), std::end(vs), v) - std::begin(vs);
}
template<typename Vs>
constexpr void concat(Vs& vs1, const Vs vs2)
{
    std::copy(std::begin(vs2), std::end(vs2), std::back_inserter(vs1));
}
template<typename Vs>
constexpr Vs concatted(Vs vs1, const Vs& vs2)
{
    concat(vs1, vs2);
    return vs1;
}
template<typename Vs, typename V>
constexpr void fillAll(Vs& arr, const V& v)
{
    if constexpr (std::is_convertible<V, Vs>::value) {
        arr = v;
    } else {
        for (auto& subarr : arr) { fillAll(subarr, v); }
    }
}
template<typename T, typename F>
constexpr Vec<T> genVec(int n, F gen)
{
    Vec<T> ans;
    std::generate_n(std::back_inserter(ans), n, gen);
    return ans;
}
template<typename Vs>
constexpr auto minAll(const Vs& vs)
{
    return *std::min_element(std::begin(vs), std::end(vs));
}
template<typename Vs>
constexpr auto maxInd(const Vs& vs)
{
    return *std::max_element(std::begin(vs), std::end(vs));
}
template<typename Vs>
constexpr int minInd(const Vs& vs)
{
    return std::min_element(std::begin(vs), std::end(vs)) - std::begin(vs);
}
template<typename Vs>
constexpr int maxInd(const Vs& vs)
{
    return std::max_element(std::begin(vs), std::end(vs)) - std::begin(vs);
}
template<typename T = int>
constexpr Vec<T> iotaVec(int n, T offset = 0)
{
    Vec<T> ans(n);
    std::iota(std::begin(ans), std::end(ans), offset);
    return ans;
}
template<typename Vs, typename... Args>
constexpr Vec<int> iotaSort(const Vs& vs, Args... args)
{
    auto is = iotaVec(vs.size());
    std::sort(std::begin(is), std::end(is), args...);
    return is;
}
inline Vec<int> permInv(const Vec<int>& is)
{
    auto ris = is;
    for (int i = 0; i < (int)is.size(); i++) { ris[is[i]] = i; }
    return ris;
}
template<typename Vs, typename V>
constexpr void plusAll(Vs& vs, const V& v)
{
    for (auto& v_ : vs) { v_ += v; }
}
template<typename Vs>
constexpr void reverseAll(Vs& vs)
{
    std::reverse(std::begin(vs), std::end(vs));
}
template<typename Vs>
constexpr Vs reversed(Vs vs)
{
    reverseAll(vs);
    return vs;
}
template<typename Vs, typename... Args>
constexpr void sortAll(Vs& vs, Args... args)
{
    std::sort(std::begin(vs), std::end(vs), args...);
}
template<typename Vs, typename... Args>
constexpr Vs sorted(Vs vs, Args... args)
{
    sortAll(vs, args...);
    return vs;
}
inline Ostream& operator<<(Ostream& os, i128 v)
{
    bool minus = false;
    if (v < 0) { minus = true, v = -v; }
    Str ans;
    if (v == 0) { ans = "0"; }
    while (v) { ans.push_back('0' + v % 10), v /= 10; }
    std::reverse(ans.begin(), ans.end());
    return os << (minus ? "-" : "") << ans;
}
inline Ostream& operator<<(Ostream& os, u128 v)
{
    Str ans;
    if (v == 0) { ans = "0"; }
    while (v) { ans.push_back('0' + v % 10), v /= 10; }
    std::reverse(ans.begin(), ans.end());
    return os << ans;
}
constexpr bool isBitOn(u64 mask, int ind) { return (mask >> ind) & 1_u64; }
constexpr bool isBitOff(u64 mask, int ind) { return not isBitOn(mask, ind); }
constexpr int topBit(u64 v) { return v == 0 ? -1 : 63 - __builtin_clzll(v); }
constexpr int lowBit(u64 v) { return v == 0 ? 64 : __builtin_ctzll(v); }
constexpr int bitWidth(u64 v) { return topBit(v) + 1; }
constexpr u64 bitCeil(u64 v) { return v ? (1_u64 << bitWidth(v - 1)) : 1_u64; }
constexpr u64 bitFloor(u64 v) { return v ? (1_u64 << topBit(v)) : 0_u64; }
constexpr bool hasSingleBit(u64 v) { return (v > 0) and ((v & (v - 1)) == 0); }
constexpr u64 bitMask(int bitWidth) { return (bitWidth == 64 ? ~0_u64 : (1_u64 << bitWidth) - 1); }
constexpr u64 bitMask(int start, int end) { return bitMask(end - start) << start; }
constexpr int popCount(u64 v) { return v ? __builtin_popcountll(v) : 0; }
constexpr bool popParity(u64 v) { return v > 0 and __builtin_parity(v) == 1; }
template<typename F>
struct Fix : F
{
    constexpr Fix(F&& f) : F{std::forward<F>(f)} {}
    template<typename... Args>
    constexpr auto operator()(Args&&... args) const
    {
        return F::operator()(*this, std::forward<Args>(args)...);
    }
};
class irange
{
private:
    struct itr
    {
        constexpr itr(i64 start = 0, i64 step = 1) : m_cnt{start}, m_step{step} {}
        constexpr bool operator!=(const itr& it) const { return m_cnt != it.m_cnt; }
        constexpr i64 operator*() { return m_cnt; }
        constexpr itr& operator++() { return m_cnt += m_step, *this; }
        i64 m_cnt, m_step;
    };
    i64 m_start, m_end, m_step;
public:
    static constexpr i64 cnt(i64 start, i64 end, i64 step)
    {
        if (step == 0) { return -1; }
        const i64 d = (step > 0 ? step : -step);
        const i64 l = (step > 0 ? start : end);
        const i64 r = (step > 0 ? end : start);
        i64 n = (r - l) / d + ((r - l) % d ? 1 : 0);
        if (l >= r) { n = 0; }
        return n;
    }
    constexpr irange(i64 start, i64 end, i64 step = 1)
        : m_start{start}, m_end{m_start + step * cnt(start, end, step)}, m_step{step}
    {
        assert(step != 0);
    }
    constexpr itr begin() const { return itr{m_start, m_step}; }
    constexpr itr end() const { return itr{m_end, m_step}; }
};
constexpr irange rep(i64 end) { return irange(0, end, 1); }
constexpr irange per(i64 rend) { return irange(rend - 1, -1, -1); }
class Scanner
{
public:
    Scanner(Istream& is = std::cin) : m_is{is} { m_is.tie(nullptr)->sync_with_stdio(false); }
    template<typename T>
    T val()
    {
        T v;
        return m_is >> v, v;
    }
    template<typename T>
    T val(T offset)
    {
        return val<T>() - offset;
    }
    template<typename T>
    Vec<T> vec(int n)
    {
        return genVec<T>(n, [&]() { return val<T>(); });
    }
    template<typename T>
    Vec<T> vec(int n, T offset)
    {
        return genVec<T>(n, [&]() { return val<T>(offset); });
    }
    template<typename T>
    Vec<Vec<T>> vvec(int n, int m)
    {
        return genVec<Vec<T>>(n, [&]() { return vec<T>(m); });
    }
    template<typename T>
    Vec<Vec<T>> vvec(int n, int m, const T offset)
    {
        return genVec<Vec<T>>(n, [&]() { return vec<T>(m, offset); });
    }
    template<typename... Args>
    auto tup()
    {
        return Tup<Args...>{val<Args>()...};
    }
    template<typename... Args>
    auto tup(Args... offsets)
    {
        return Tup<Args...>{val<Args>(offsets)...};
    }
private:
    Istream& m_is;
};
inline Scanner in;
class Printer
{
public:
    Printer(Ostream& os = std::cout) : m_os{os} { m_os << std::fixed << std::setprecision(15); }
    template<typename... Args>
    int operator()(const Args&... args)
    {
        return dump(args...), 0;
    }
    template<typename... Args>
    int ln(const Args&... args)
    {
        return dump(args...), m_os << '\n', 0;
    }
    template<typename... Args>
    int el(const Args&... args)
    {
        return dump(args...), m_os << std::endl, 0;
    }
    int YES(bool b = true) { return ln(b ? "YES" : "NO"); }
    int NO(bool b = true) { return YES(not b); }
    int Yes(bool b = true) { return ln(b ? "Yes" : "No"); }
    int No(bool b = true) { return Yes(not b); }
private:
    template<typename T>
    void dump(const T& v)
    {
        m_os << v;
    }
    template<typename T>
    void dump(const Vec<T>& vs)
    {
        for (int i : rep(vs.size())) { m_os << (i ? " " : ""), dump(vs[i]); }
    }
    template<typename T>
    void dump(const Vec<Vec<T>>& vss)
    {
        for (int i : rep(vss.size())) { m_os << (i ? "\n" : ""), dump(vss[i]); }
    }
    template<typename T, typename... Ts>
    int dump(const T& v, const Ts&... args)
    {
        return dump(v), m_os << ' ', dump(args...), 0;
    }
    Ostream& m_os;
};
inline Printer out;
template<typename T, int n, int i = 0>
auto ndVec(int const (&szs)[n], const T x = T{})
{
    if constexpr (i == n) {
        return x;
    } else {
        return std::vector(szs[i], ndVec<T, n, i + 1>(szs, x));
    }
}
template<typename T, typename F>
inline T binSearch(T ng, T ok, F check)
{
    while (std::abs(ok - ng) > 1) {
        const T mid = (ok + ng) / 2;
        (check(mid) ? ok : ng) = mid;
    }
    return ok;
}
template<typename T, typename F>
inline T fbinSearch(T ng, T ok, F check, int times)
{
    for (auto _temp_name_0 [[maybe_unused]] : rep(times)) {
        const T mid = (ok + ng) / 2;
        (check(mid) ? ok : ng) = mid;
    }
    return ok;
}
template<typename T>
constexpr T clampAdd(T x, T y, T min, T max)
{
    return std::clamp(x + y, min, max);
}
template<typename T>
constexpr T clampSub(T x, T y, T min, T max)
{
    return std::clamp(x - y, min, max);
}
template<typename T>
constexpr T clampMul(T x, T y, T min, T max)
{
    if (y < 0) { x = -x, y = -y; }
    const T xmin = ceilDiv(min, y);
    const T xmax = floorDiv(max, y);
    if (x < xmin) {
        return min;
    } else if (x > xmax) {
        return max;
    } else {
        return x * y;
    }
}
template<typename T>
constexpr T clampDiv(T x, T y, T min, T max)
{
    return std::clamp(floorDiv(x, y), min, max);
}
template<typename T>
constexpr Pair<T, T> extgcd(const T a, const T b) // [x,y] -> ax+by=gcd(a,b)
{
    static_assert(std::is_signed_v<T>, "Signed integer is allowed.");
    assert(a != 0 or b != 0);
    if (a >= 0 and b >= 0) {
        if (a < b) {
            const auto [y, x] = extgcd(b, a);
            return {x, y};
        }
        if (b == 0) { return {1, 0}; }
        const auto [x, y] = extgcd(b, a % b);
        return {y, x - (a / b) * y};
    } else {
        auto [x, y] = extgcd(std::abs(a), std::abs(b));
        if (a < 0) { x = -x; }
        if (b < 0) { y = -y; }
        return {x, y};
    }
}
template<typename T>
constexpr T inverse(const T a, const T mod) // ax=gcd(a,M) (mod M)
{
    assert(a > 0 and mod > 0);
    auto [x, y] = extgcd(a, mod);
    if (x <= 0) { x += mod; }
    return x;
}
template<u32 mod_, u32 root_, u32 max2p_>
class modint
{
    template<typename U = u32&>
    static U modRef()
    {
        static u32 s_mod = 0;
        return s_mod;
    }
    template<typename U = u32&>
    static U rootRef()
    {
        static u32 s_root = 0;
        return s_root;
    }
    template<typename U = u32&>
    static U max2pRef()
    {
        static u32 s_max2p = 0;
        return s_max2p;
    }
public:
    static_assert(mod_ <= LIMMAX<i32>, "mod(signed int size) only supported!");
    static constexpr bool isDynamic() { return (mod_ == 0); }
    template<typename U = const u32>
    static constexpr std::enable_if_t<mod_ != 0, U> mod()
    {
        return mod_;
    }
    template<typename U = const u32>
    static std::enable_if_t<mod_ == 0, U> mod()
    {
        return modRef();
    }
    template<typename U = const u32>
    static constexpr std::enable_if_t<mod_ != 0, U> root()
    {
        return root_;
    }
    template<typename U = const u32>
    static std::enable_if_t<mod_ == 0, U> root()
    {
        return rootRef();
    }
    template<typename U = const u32>
    static constexpr std::enable_if_t<mod_ != 0, U> max2p()
    {
        return max2p_;
    }
    template<typename U = const u32>
    static std::enable_if_t<mod_ == 0, U> max2p()
    {
        return max2pRef();
    }
    template<typename U = u32>
    static void setMod(std::enable_if_t<mod_ == 0, U> m)
    {
        assert(1 <= m and m <= LIMMAX<i32>);
        modRef() = m;
        sinvRef() = {1, 1};
        factRef() = {1, 1};
        ifactRef() = {1, 1};
    }
    template<typename U = u32>
    static void setRoot(std::enable_if_t<mod_ == 0, U> r)
    {
        rootRef() = r;
    }
    template<typename U = u32>
    static void setMax2p(std::enable_if_t<mod_ == 0, U> m)
    {
        max2pRef() = m;
    }
    constexpr modint() : m_val{0} {}
    constexpr modint(i64 v) : m_val{normll(v)} {}
    constexpr void setRaw(u32 v) { m_val = v; }
    constexpr modint operator-() const { return modint{0} - (*this); }
    constexpr modint& operator+=(const modint& m)
    {
        m_val = norm(m_val + m.val());
        return *this;
    }
    constexpr modint& operator-=(const modint& m)
    {
        m_val = norm(m_val + mod() - m.val());
        return *this;
    }
    constexpr modint& operator*=(const modint& m)
    {
        m_val = normll((i64)m_val * (i64)m.val() % (i64)mod());
        return *this;
    }
    constexpr modint& operator/=(const modint& m) { return *this *= m.inv(); }
    constexpr modint operator+(const modint& m) const
    {
        auto v = *this;
        return v += m;
    }
    constexpr modint operator-(const modint& m) const
    {
        auto v = *this;
        return v -= m;
    }
    constexpr modint operator*(const modint& m) const
    {
        auto v = *this;
        return v *= m;
    }
    constexpr modint operator/(const modint& m) const
    {
        auto v = *this;
        return v /= m;
    }
    constexpr bool operator==(const modint& m) const { return m_val == m.val(); }
    constexpr bool operator!=(const modint& m) const { return not(*this == m); }
    friend Istream& operator>>(Istream& is, modint& m)
    {
        i64 v;
        return is >> v, m = v, is;
    }
    friend Ostream& operator<<(Ostream& os, const modint& m) { return os << m.val(); }
    constexpr u32 val() const { return m_val; }
    template<typename I>
    constexpr modint pow(I n) const
    {
        return powerInt(*this, n);
    }
    constexpr modint inv() const { return inverse<i32>(m_val, mod()); }
    static modint sinv(u32 n)
    {
        auto& is = sinvRef();
        for (u32 i = (u32)is.size(); i <= n; i++) { is.push_back(-is[mod() % i] * (mod() / i)); }
        return is[n];
    }
    static modint fact(u32 n)
    {
        auto& fs = factRef();
        for (u32 i = (u32)fs.size(); i <= n; i++) { fs.push_back(fs.back() * i); }
        return fs[n];
    }
    static modint ifact(u32 n)
    {
        auto& ifs = ifactRef();
        for (u32 i = (u32)ifs.size(); i <= n; i++) { ifs.push_back(ifs.back() * sinv(i)); }
        return ifs[n];
    }
    static modint perm(int n, int k) { return k > n or k < 0 ? modint{0} : fact(n) * ifact(n - k); }
    static modint comb(int n, int k)
    {
        return k > n or k < 0 ? modint{0} : fact(n) * ifact(n - k) * ifact(k);
    }
private:
    static Vec<modint>& sinvRef()
    {
        static Vec<modint> is{1, 1};
        return is;
    }
    static Vec<modint>& factRef()
    {
        static Vec<modint> fs{1, 1};
        return fs;
    }
    static Vec<modint>& ifactRef()
    {
        static Vec<modint> ifs{1, 1};
        return ifs;
    }
    static constexpr u32 norm(u32 x) { return x < mod() ? x : x - mod(); }
    static constexpr u32 normll(i64 x) { return norm(u32(x % (i64)mod() + (i64)mod())); }
    u32 m_val;
};
using modint_1000000007 = modint<1000000007, 5, 1>;
using modint_998244353 = modint<998244353, 3, 23>;
template<int id>
using modint_dynamic = modint<0, 0, id>;
template<typename T = int>
class Graph
{
    struct Edge
    {
        Edge() = default;
        Edge(int i, int t, T c) : id{i}, to{t}, cost{c} {}
        int id;
        int to;
        T cost;
        operator int() const { return to; }
    };
public:
    Graph(int n) : m_v{n}, m_edges(n) {}
    void addEdge(int u, int v, bool bi = false)
    {
        assert(0 <= u and u < m_v);
        assert(0 <= v and v < m_v);
        m_edges[u].emplace_back(m_e, v, 1);
        if (bi) { m_edges[v].emplace_back(m_e, u, 1); }
        m_e++;
    }
    void addEdge(int u, int v, const T& c, bool bi = false)
    {
        assert(0 <= u and u < m_v);
        assert(0 <= v and v < m_v);
        m_edges[u].emplace_back(m_e, v, c);
        if (bi) { m_edges[v].emplace_back(m_e, u, c); }
        m_e++;
    }
    const Vec<Edge>& operator[](const int u) const
    {
        assert(0 <= u and u < m_v);
        return m_edges[u];
    }
    Vec<Edge>& operator[](const int u)
    {
        assert(0 <= u and u < m_v);
        return m_edges[u];
    }
    int v() const { return m_v; }
    int e() const { return m_e; }
    friend Ostream& operator<<(Ostream& os, const Graph& g)
    {
        for (int u : rep(g.v())) {
            for (const auto& [id, v, c] : g[u]) {
                os << "[" << id << "]: ";
                os << u << "->" << v << "(" << c << ")\n";
            }
        }
        return os;
    }
    Vec<T> sizes(int root = 0) const
    {
        const int N = v();
        assert(0 <= root and root < N);
        Vec<T> ss(N, 1);
        Fix([&](auto dfs, int u, int p) -> void {
            for ([[maybe_unused]] const auto& [_temp_name_1, v, c] : m_edges[u]) {
                if (v == p) { continue; }
                dfs(v, u);
                ss[u] += ss[v];
            }
        })(root, -1);
        return ss;
    }
    Vec<T> depths(int root = 0) const
    {
        const int N = v();
        assert(0 <= root and root < N);
        Vec<T> ds(N, 0);
        Fix([&](auto dfs, int u, int p) -> void {
            for ([[maybe_unused]] const auto& [_temp_name_2, v, c] : m_edges[u]) {
                if (v == p) { continue; }
                ds[v] = ds[u] + c;
                dfs(v, u);
            }
        })(root, -1);
        return ds;
    }
    Vec<int> parents(int root = 0) const
    {
        const int N = v();
        assert(0 <= root and root < N);
        Vec<int> ps(N, -1);
        Fix([&](auto dfs, int u, int p) -> void {
            for ([[maybe_unused]] const auto& [_temp_name_3, v, c] : m_edges[u]) {
                if (v == p) { continue; }
                ps[v] = u;
                dfs(v, u);
            }
        })(root, -1);
        return ps;
    }
private:
    int m_v;
    int m_e = 0;
    Vec<Vec<Edge>> m_edges;
};
template<typename Engine>
class RNG
{
public:
    using result_type = typename Engine::result_type;
    using T = result_type;
    static constexpr T min() { return Engine::min(); }
    static constexpr T max() { return Engine::max(); }
    RNG() : RNG(std::random_device{}()) {}
    RNG(T seed) : m_rng(seed) {}
    T operator()() { return m_rng(); }
    template<typename T>
    T val(T min, T max)
    {
        return std::uniform_int_distribution<T>(min, max)(m_rng);
    }
    template<typename T, typename... Args>
    auto tup(T min, T max, const Args&... offsets)
    {
        return Tup<T, Args...>{val<T>(min, max), val<Args>(offsets)...};
    }
    template<typename T>
    Vec<T> vec(int n, T min, T max)
    {
        return genVec<T>(n, [&]() { return val<T>(min, max); });
    }
    template<typename T>
    Vec<Vec<T>> vvec(int n, int m, T min, T max)
    {
        return genVec<Vec<T>>(n, [&]() { return vec(m, min, max); });
    }
private:
    Engine m_rng;
};
inline RNG<std::mt19937> rng;
inline RNG<std::mt19937_64> rng64;
template<u64 mod_, u64 root_, u64 max2p_>
class modint64
{
    template<typename U = u64&>
    static U modRef()
    {
        static u64 s_mod = 0;
        return s_mod;
    }
    template<typename U = u64&>
    static U rootRef()
    {
        static u64 s_root = 0;
        return s_root;
    }
    template<typename U = u64&>
    static U max2pRef()
    {
        static u64 s_max2p = 0;
        return s_max2p;
    }
public:
    static_assert(mod_ <= LIMMAX<i64>, "mod(signed int size) only supported!");
    static constexpr bool isDynamic() { return (mod_ == 0); }
    template<typename U = const u64>
    static constexpr std::enable_if_t<mod_ != 0, U> mod()
    {
        return mod_;
    }
    template<typename U = const u64>
    static std::enable_if_t<mod_ == 0, U> mod()
    {
        return modRef();
    }
    template<typename U = const u64>
    static constexpr std::enable_if_t<mod_ != 0, U> root()
    {
        return root_;
    }
    template<typename U = const u64>
    static std::enable_if_t<mod_ == 0, U> root()
    {
        return rootRef();
    }
    template<typename U = const u64>
    static constexpr std::enable_if_t<mod_ != 0, U> max2p()
    {
        return max2p_;
    }
    template<typename U = const u64>
    static std::enable_if_t<mod_ == 0, U> max2p()
    {
        return max2pRef();
    }
    template<typename U = u64>
    static void setMod(std::enable_if_t<mod_ == 0, U> m)
    {
        assert(1 <= m and m <= LIMMAX<i64>);
        modRef() = m;
        sinvRef() = {1, 1};
        factRef() = {1, 1};
        ifactRef() = {1, 1};
    }
    template<typename U = u64>
    static void setRoot(std::enable_if_t<mod_ == 0, U> r)
    {
        rootRef() = r;
    }
    template<typename U = u64>
    static void setMax2p(std::enable_if_t<mod_ == 0, U> m)
    {
        max2pRef() = m;
    }
    constexpr modint64() : m_val{0} {}
    constexpr modint64(const i64 v) : m_val{normLL(v)} {}
    constexpr void setRaw(const u64 v) { m_val = v; }
    constexpr modint64 operator+() const { return *this; }
    constexpr modint64 operator-() const { return modint64{0} - (*this); }
    constexpr modint64& operator+=(const modint64& m)
    {
        m_val = norm(m_val + m.val());
        return *this;
    }
    constexpr modint64& operator-=(const modint64& m)
    {
        m_val = norm(m_val + mod() - m.val());
        return *this;
    }
    constexpr modint64& operator*=(const modint64& m)
    {
        m_val = normLL((i128)m_val * (i128)m.val() % (i128)mod());
        return *this;
    }
    constexpr modint64& operator/=(const modint64& m) { return *this *= m.inv(); }
    constexpr modint64 operator+(const modint64& m) const
    {
        auto v = *this;
        return v += m;
    }
    constexpr modint64 operator-(const modint64& m) const
    {
        auto v = *this;
        return v -= m;
    }
    constexpr modint64 operator*(const modint64& m) const
    {
        auto v = *this;
        return v *= m;
    }
    constexpr modint64 operator/(const modint64& m) const
    {
        auto v = *this;
        return v /= m;
    }
    constexpr bool operator==(const modint64& m) const { return m_val == m.val(); }
    constexpr bool operator!=(const modint64& m) const { return not(*this == m); }
    friend Istream& operator>>(Istream& is, modint64& m)
    {
        i64 v;
        return is >> v, m = v, is;
    }
    friend Ostream& operator<<(Ostream& os, const modint64& m) { return os << m.val(); }
    constexpr u64 val() const { return m_val; }
    template<typename I>
    constexpr modint64 pow(I n) const
    {
        return powerInt(*this, n);
    }
    constexpr modint64 inv() const { return inverse<i64>(m_val, mod()); }
    modint64 sinv() const { return sinv(m_val); }
    static modint64 sinv(u32 n)
    {
        auto& is = sinvRef();
        for (u32 i = (u32)is.size(); i <= n; i++) { is.push_back(-is[mod() % i] * (mod() / i)); }
        return is[n];
    }
    static modint64 fact(u32 n)
    {
        auto& fs = factRef();
        for (u32 i = (u32)fs.size(); i <= n; i++) { fs.push_back(fs.back() * i); }
        return fs[n];
    }
    static modint64 ifact(u32 n)
    {
        auto& ifs = ifactRef();
        for (u32 i = (u32)ifs.size(); i <= n; i++) { ifs.push_back(ifs.back() * sinv(i)); }
        return ifs[n];
    }
    static modint64 perm(int n, int k)
    {
        return k > n or k < 0 ? modint64{0} : fact(n) * ifact(n - k);
    }
    static modint64 comb(int n, int k)
    {
        return k > n or k < 0 ? modint64{0} : fact(n) * ifact(n - k) * ifact(k);
    }
private:
    static Vec<modint64>& sinvRef()
    {
        static Vec<modint64> is{1, 1};
        return is;
    }
    static Vec<modint64>& factRef()
    {
        static Vec<modint64> fs{1, 1};
        return fs;
    }
    static Vec<modint64>& ifactRef()
    {
        static Vec<modint64> ifs{1, 1};
        return ifs;
    }
    static constexpr u64 norm(const u64 x) { return x < mod() ? x : x - mod(); }
    static constexpr u64 normLL(const i64 x)
    {
        return norm(u64((i128)x % (i128)mod() + (i128)mod()));
    }
    u64 m_val;
};
template<int id>
using modint64_dynamic = modint64<0, 0, id>;
template<typename mint>
bool millerRabin(u64 n, const Vec<u64>& as)
{
    auto d = n - 1;
    for (; (d & 1) == 0; d >>= 1) {}
    for (const u64 a : as) {
        if (n <= a) { break; }
        auto s = d;
        mint x = mint(a).pow(s);
        while (x.val() != 1 and x.val() != n - 1 and s != n - 1) { x *= x, s <<= 1; }
        if (x.val() != n - 1 and s % 2 == 0) { return false; }
    }
    return true;
}
inline bool isPrime(u64 n)
{
    using mint = modint_dynamic<873293817>;
    using mint64 = modint64_dynamic<828271328>;
    if (n == 1) { return false; }
    if ((n & 1) == 0) { return n == 2; }
    if (n < (1ULL << 30)) {
        mint::setMod(n);
        return millerRabin<mint>(n, {2, 7, 61});
    } else {
        mint64::setMod(n);
        return millerRabin<mint64>(n, {2, 325, 9375, 28178, 450775, 9780504});
    }
}
template<typename mint>
u64 pollardRho(u64 n)
{
    if (n % 2 == 0) { return 2; }
    if (isPrime(n)) { return n; }
    mint c;
    auto f = [&](const mint& x) { return x * x + c; };
    while (true) {
        mint x, y, ys, q = 1;
        y = rng64.val<u64>(0, n - 2) + 2;
        c = rng64.val<u64>(0, n - 2) + 2;
        u64 d = 1;
        constexpr u32 dk = 128;
        for (u32 r = 1; d == 1; r <<= 1) {
            x = y;
            for (u32 i = 0; i < r; i++) { y = f(y); }
            for (u32 k = 0; k < r and d == 1; k += dk) {
                ys = y;
                for (u32 i = 0; i < dk and i < r - k; i++) { q *= x - (y = f(y)); }
                d = std::gcd((u64)q.val(), n);
            }
        }
        if (d == n) {
            do {
                d = std::gcd(u64((x - (ys = f(ys))).val()), n);
            } while (d == 1);
        }
        if (d != n) { return d; }
    }
    return n;
}
Map<u64, int> primeFactors(u64 n)
{
    using mint = modint_dynamic<287687412>;
    using mint64 = modint64_dynamic<4832432>;
    Map<u64, int> ans;
    Fix([&](auto dfs, u64 x) -> void {
        while ((x & 1) == 0) { x >>= 1, ans[2]++; }
        if (x == 1) { return; }
        u64 p;
        if (x < (1ULL << 30)) {
            mint::setMod(x);
            p = pollardRho<mint>(x);
        } else {
            mint64::setMod(x);
            p = pollardRho<mint64>(x);
        }
        if (p == x) {
            ans[p]++;
            return;
        }
        dfs(p), dfs(x / p);
    })(n);
    return ans;
}
Vec<u64> divisors(const u64 n)
{
    const auto fs = primeFactors(n);
    Vec<u64> ds{1};
    for (const auto& [p, e] : fs) {
        u64 pe = p;
        const u32 dn = ds.size();
        for (i32 i = 0; i < e; i++, pe *= p) {
            for (u32 j = 0; j < dn; j++) { ds.push_back(ds[j] * pe); }
        }
    }
    return ds;
}
template<typename mint>
class NumberTheoriticTransform
{
    // DynamicModint 非対応
    static_assert(not mint::isDynamic(), "class NTT: Not support dynamic modint.");
private:
    static constexpr u32 MOD = mint::mod();
    static constexpr u32 ROOT = mint::root();
    static constexpr u32 L_MAX = mint::max2p();
    static constexpr int N_MAX = (1 << L_MAX);
public:
    static Vec<mint> convolute(Vec<mint> as, Vec<mint> bs)
    {
        const int AN = as.size();
        const int BN = bs.size();
        const int CN = AN + BN - 1;
        const int N = bitCeil(CN);
        as.resize(N, 0), bs.resize(N, 0);
        transform(as, false), transform(bs, false);
        for (int i : rep(N)) { as[i] *= bs[i]; }
        transform(as, true);
        as.resize(CN);
        return as;
    }
    static void transform(Vec<mint>& as, bool rev)
    {
        const int N = as.size();
        assert(hasSingleBit(N));
        if (N == 1) { return; }
        const int L = topBit(N);
        const auto l_range = (rev ? irange(1, L + 1, 1) : irange(L, 0, -1));
        for (int l : l_range) {
            const int H = 1 << l;
            const int B = N / H;
            for (int b : rep(B)) {
                const mint W = zeta(l, rev);
                mint W_h = 1;
                for (int h : rep(H / 2)) {
                    const int y1 = H * b + h;
                    const int y2 = y1 + H / 2;
                    const mint a1 = as[y1];
                    const mint a2 = as[y2];
                    const mint na1 = (rev ? a1 + a2 * W_h : a1 + a2);
                    const mint na2 = (rev ? a1 - a2 * W_h : (a1 - a2) * W_h);
                    as[y1] = na1;
                    as[y2] = na2;
                    W_h *= W;
                }
            }
        }
        if (rev) {
            const mint iN = mint::sinv(N);
            for (auto& a : as) { a *= iN; }
        }
    }
private:
    static mint zeta(int i, bool rev)
    {
        static Vec<mint> zs; // zs[i] = 1の2^i乗根
        static Vec<mint> izs; // izs[i] = zs[i]の逆元
        if (zs.empty()) {
            zs.resize(L_MAX + 1, 1);
            izs.resize(L_MAX + 1, 1);
            zs[L_MAX] = mint(ROOT).pow((MOD - 1) / (1 << L_MAX));
            izs[L_MAX] = zs[L_MAX].inv();
            for (int l : per(L_MAX)) {
                zs[l] = zs[l + 1] * zs[l + 1];
                izs[l] = izs[l + 1] * izs[l + 1];
            }
        }
        return (rev ? izs[i] : zs[i]);
    }
};
class Garner
{
public:
    template<typename mint, typename mint1, typename mint2>
    static mint restore_mod(const mint1& x1, const mint2& x2)
    {
        constexpr auto m1 = mint1::mod();
        const auto [y0, y1] = coeff(x1, x2);
        return mint(y0.val()) + mint(y1.val()) * m1;
    }
    template<typename mint, typename mint1, typename mint2, typename mint3>
    static mint restore_mod(const mint1& x1, const mint2& x2, const mint3& x3)
    {
        constexpr auto m1 = mint1::mod();
        constexpr auto m2 = mint2::mod();
        const auto [y0, y1, y2] = coeff(x1, x2, x3);
        return mint(y0.val()) + mint(y1.val()) * m1 + mint(y2.val()) * m1 * m2;
    }
    template<typename mint1, typename mint2>
    static i64 restore_i64(const mint1& x1, const mint2& x2)
    {
        constexpr u32 m1 = mint1::mod();
        constexpr u32 m2 = mint2::mod();
        const auto [y0, y1] = coeff(x1, x2);
        constexpr u64 MAX = 1_u64 << 63;
        const i128 M = (i128)m1 * m2;
        i128 S = i128(y0.val()) + i128(y1.val()) * m1;
        if (S >= MAX) { S -= M; }
        return (i64)S;
    }
    template<typename mint1, typename mint2, typename mint3>
    static i64 restore_i64(const mint1& x1, const mint2& x2, const mint3& x3)
    {
        constexpr u32 m1 = mint1::mod();
        constexpr u32 m2 = mint2::mod();
        constexpr u32 m3 = mint3::mod();
        const auto [y0, y1, y2] = coeff(x1, x2, x3);
        constexpr u64 MAX = 1_u64 << 63;
        const i128 M = (i128)m1 * m2 * m3;
        i128 S = i128(y0.val()) + i128(y1.val()) * m1 + i128(y2.val()) * m1 * m2;
        if (S >= MAX) { S -= M; }
        return (i64)S;
    }
private:
    template<typename mint1, typename mint2>
    static Pair<mint1, mint2> coeff(const mint1& x1, const mint2& x2)
    {
        constexpr auto m1 = mint1::mod();
        constexpr mint2 m1_inv = mint2(m1).inv();
        const mint1 y0 = x1;
        const mint2 y1 = (x2 - mint2(y0.val())) * m1_inv;
        return {y0, y1};
    }
    template<typename mint1, typename mint2, typename mint3>
    static Tup<mint1, mint2, mint3> coeff(const mint1& x1, const mint2& x2, const mint3& x3)
    {
        constexpr auto m1 = mint1::mod();
        constexpr auto m2 = mint2::mod();
        constexpr mint2 m1_inv = mint2(m1).inv();
        constexpr mint3 m1m2_inv = (mint3(m1) * mint3(m2)).inv();
        const mint1 y0 = x1;
        const mint2 y1 = (x2 - mint2(y0.val())) * m1_inv;
        const mint3 y2 = (x3 - mint3(y0.val()) - mint3(y1.val()) * m1) * m1m2_inv;
        return {y0, y1, y2};
    }
};
template<typename mint>
Vec<mint> convolute_mod(const Vec<mint>& as, const Vec<mint>& bs)
{
    constexpr u32 L_MAX = mint::max2p();
    constexpr int N_MAX = (1 << L_MAX);
    const int AN = as.size();
    const int BN = bs.size();
    if (AN == 0 or BN == 0) { return {}; }
    if (AN > BN) { return convolute_mod(bs, as); }
    const int N = AN + BN - 1;
    if (AN * 10 <= BN) {
        Vec<mint> cs(N, 0);
        for (int sj : irange(0, BN, AN)) {
            const int tj = std::min(BN, sj + AN);
            const auto bbs = Vec<mint>(std::begin(bs) + sj, std::begin(bs) + tj);
            const auto bcs = convolute_mod(as, bbs);
            for (int dj : rep(bcs.size())) { cs[sj + dj] += bcs[dj]; }
        }
        return cs;
    }
    if (N <= N_MAX) {
        // mintはNTT Friendlyなのでそのまま畳み込み
        return NumberTheoriticTransform<mint>::convolute(as, bs);
    } else {
        assert(N <= (1 << 24));
        using submint1 = modint<469762049, 3, 26>;
        using submint2 = modint<167772161, 3, 25>;
        using submint3 = modint<754974721, 11, 24>;
        // mod 3つでGarner復元
        Vec<submint1> as1(AN), bs1(BN);
        Vec<submint2> as2(AN), bs2(BN);
        Vec<submint3> as3(AN), bs3(BN);
        for (int i : rep(AN)) { as1[i] = as[i].val(), as2[i] = as[i].val(), as3[i] = as[i].val(); }
        for (int i : rep(BN)) { bs1[i] = bs[i].val(), bs2[i] = bs[i].val(), bs3[i] = bs[i].val(); }
        const auto cs1 = NumberTheoriticTransform<submint1>::convolute(as1, bs1);
        const auto cs2 = NumberTheoriticTransform<submint2>::convolute(as2, bs2);
        const auto cs3 = NumberTheoriticTransform<submint3>::convolute(as3, bs3);
        Vec<mint> cs(N);
        for (int i : rep(N)) { cs[i] = Garner::restore_mod<mint>(cs1[i], cs2[i], cs3[i]); }
        return cs;
    }
}
template<typename I>
Vec<i64> convolute_i64(const Vec<I>& as, const Vec<I>& bs)
{
    const int AN = as.size();
    const int BN = bs.size();
    if (AN == 0 or BN == 0) { return {}; }
    if (AN > BN) { return convolute_i64<I>(bs, as); }
    const int N = AN + BN - 1;
    assert(N <= (1 << 24));
    if (AN * 2 <= BN) {
        Vec<i64> cs(N, 0);
        for (int sj : irange(0, BN, AN)) {
            const int tj = std::min(BN, sj + AN);
            const auto bbs = Vec<I>(std::begin(bs) + sj, std::begin(bs) + tj);
            const auto bcs = convolute_i64<I>(as, bbs);
            for (int dj : rep(bcs.size())) { cs[sj + dj] += bcs[dj]; }
        }
        return cs;
    }
    using submint1 = modint<469762049, 3, 26>;
    using submint2 = modint<167772161, 3, 25>;
    using submint3 = modint<754974721, 11, 24>;
    // mod 3つでGarner復元
    Vec<submint1> as1(AN), bs1(BN);
    Vec<submint2> as2(AN), bs2(BN);
    Vec<submint3> as3(AN), bs3(BN);
    for (int i : rep(AN)) { as1[i] = as[i], as2[i] = as[i], as3[i] = as[i]; }
    for (int i : rep(BN)) { bs1[i] = bs[i], bs2[i] = bs[i], bs3[i] = bs[i]; }
    const auto cs1 = NumberTheoriticTransform<submint1>::convolute(as1, bs1);
    const auto cs2 = NumberTheoriticTransform<submint2>::convolute(as2, bs2);
    const auto cs3 = NumberTheoriticTransform<submint3>::convolute(as3, bs3);
    Vec<i64> cs(N);
    for (int i : rep(N)) { cs[i] = Garner::restore_i64(cs1[i], cs2[i], cs3[i]); }
    return cs;
}
int main()
{
    auto solve = [&]() {
        const auto [N, P] = in.tup<int, int>();
        using mintP = modint_dynamic<0>;
        mintP::setMod(P);
        using mint998244353 = modint_998244353;
        const auto As = in.vec<int>(N);
        int ZN = 0;
        for (int A : As) {
            if (A == 0) {
                ZN++;
            }
        }
        if (ZN == 0) {
            out.ln(1, 1);
            out.ln(0);
            return 0;
        }
        if (ZN == N) {
            out.ln(P, 1);
            out.ln(iotaVec(P));
            return 0;
        }
        if (P == 2) {
            out.ln(1, 2);
            out.ln(1);
            return 0;
        }
        int g = 1; // 原始根
        {
            Vec<int> ps;
            for (const auto& [p, e] : primeFactors(P - 1)) {
                ps.push_back(p);
            }
            for (g = 1; g < P; g++) {
                bool ok = true;
                for (int p : ps) {
                    if (mintP(g).pow((P - 1) / p) == 1) {
                        ok = false;
                    }
                }
                if (ok) {
                    break;
                }
            }
        }
        Vec<int> logs(P, -1); // 対数
        {
            mintP p = 1;
            for (int i : rep(P - 1)) {
                logs[p.val()] = i;
                p *= g;
            }
        }
        Vec<mint998244353> as((P - 1) * 2, 0);
        for (int A : As) {
            if (A == 0) {
                continue;
            }
            const int l = logs[A];
            as[l] = 1, as[l + (P - 1)] = 1;
        }
        reverseAll(as);
        auto check = [&](int M) { // 1 ~ M-1 が含まれるか？
            Vec<mint998244353> bs(P - 1, 0);
            for (int m : irange(1, M)) {
                bs[logs[m]] = 1;
            }
            int B = 0;
            for (int i : rep(P - 1)) {
                if (bs[i] != 0) {
                    B++;
                }
            }
            const auto cs = convolute_mod(as, bs);
            Vec<int> ks;
            for (int k : rep(P - 1)) {
                const auto c = cs[P * 2 - 3 - k];
                if (c == B) {
                    ks.push_back(k);
                }
            }
            return ks;
        };
        const int M
            = binSearch(P + 1, 1, [&](int M) { return not check(M).empty(); });
        const auto ks = check(M);
        out.ln(ks.size(), M);
        Vec<int> cs;
        for (int k : ks) {
            cs.push_back(mintP(g).pow(P - 1 - k).val());
        }
        sortAll(cs);
        out.ln(cs);
        return 0;
    };
    const auto T = in.val<int>();
    for (auto _temp_name_4 [[maybe_unused]] : rep(T)) {
        solve();
    }
    return 0;
}