bool TEST = false;

using namespace std;
#include<bits/stdc++.h>
#include<fstream>

#define rep(i,n) for(ll (i)=0;(i)<(ll)(n);i++)
#define rrep(i,n) for(ll (i)=(ll)(n)-1;(i)>=0;i--)
#define range(i,start,end,step) for(ll (i)=start;(i)<(ll)(end);(i)+=(step))
#define rrange(i,start,end,step) for(ll (i)=start;(i)>(ll)(end);(i)+=(step))

#define dump(x)  cerr << "Line " << __LINE__ << ": " <<  #x << " = " << (x) << "\n";
#define spa << " " <<
#define fi first
#define se second
#define all(a)  (a).begin(),(a).end()
#define allr(a)  (a).rbegin(),(a).rend()

using ld = long double;
using ll = long long;
using ull = unsigned long long;
using pii = pair<int, int>;
using pll = pair<ll, ll>;
using pdd = pair<ld, ld>;
 
template<typename T> using V = vector<T>;
template<typename T> using VV = V<V<T>>;
template<typename T, typename T2> using P = pair<T, T2>;
template<typename T, typename T2> using M = map<T, T2>;
template<typename T> using S = set<T>;
template<typename T, typename T2> using UM = unordered_map<T, T2>;
template<typename T> using PQ = priority_queue<T, V<T>, greater<T>>;
template<typename T> using rPQ = priority_queue<T, V<T>, less<T>>;
template<class T>vector<T> make_vec(size_t a){return vector<T>(a);}
template<class T, class... Ts>auto make_vec(size_t a, Ts... ts){return vector<decltype(make_vec<T>(ts...))>(a, make_vec<T>(ts...));}
template<class SS, class T> ostream& operator << (ostream& os, const pair<SS, T> v){os << "(" << v.first << ", " << v.second << ")"; return os;}
template<typename T> ostream& operator<<(ostream &os, const vector<T> &v) { for (auto &e : v) os << e << ' '; return os; }
template<class T> ostream& operator<<(ostream& os, const vector<vector<T>> &v){ for(auto &e : v){os << e << "\n";} return os;}
struct fast_ios { fast_ios(){ cin.tie(nullptr); ios::sync_with_stdio(false); cout << fixed << setprecision(20); }; } fast_ios_;
 
template <class T> void UNIQUE(vector<T> &x) {sort(all(x));x.erase(unique(all(x)), x.end());}
template<class T> bool chmax(T &a, const T &b) { if (a<b) { a=b; return 1; } return 0; }
template<class T> bool chmin(T &a, const T &b) { if (a>b) { a=b; return 1; } return 0; }
void fail() { cout << -1 << '\n'; exit(0); }
inline int popcount(const int x) { return __builtin_popcount(x); }
inline int popcount(const ll x) { return __builtin_popcountll(x); }
template<typename T> void debug(vector<vector<T>>&v){for(ll i=0;i<v.size();i++)
{cerr<<v[i][0];for(ll j=1;j<v[i].size();j++)cerr spa v[i][j];cerr<<"\n";}};
template<typename T> void debug(vector<T>&v){if(v.size()!=0)cerr<<v[0];
for(ll i=1;i<v.size();i++)cerr spa v[i];
cerr<<"\n";};
template<typename T> void debug(priority_queue<T>&v){V<T> vals; while(!v.empty()) {cerr << v.top() << " "; vals.push_back(v.top()); v.pop();} cerr<<"\n"; for(auto val: vals) v.push(val);}
template<typename T, typename T2> void debug(map<T,T2>&v){for(auto [k,v]: v) cerr << k spa v << "\n"; cerr<<"\n";}
template<typename T, typename T2> void debug(unordered_map<T,T2>&v){for(auto [k,v]: v) cerr << k spa v << "\n";cerr<<"\n";}
V<int> listrange(int n) {V<int> res(n); rep(i,n) res[i]=i; return res;}

template<typename T> P<T,T> divmod(T a, T b) {return make_pair(a/b, a%b);}

const ll INF = (1ll<<62);
// const ld EPS   = 1e-10;
// const ld PI    = acos(-1.0);
template< int mod >
struct ModInt {
  int x;

  ModInt() : x(0) {}

  ModInt(int64_t y) : x(y >= 0 ? y % mod : (mod - (-y) % mod) % mod) {}

  ModInt &operator+=(const ModInt &p) {
    if((x += p.x) >= mod) x -= mod;
    return *this;
  }

  ModInt &operator-=(const ModInt &p) {
    if((x += mod - p.x) >= mod) x -= mod;
    return *this;
  }

  ModInt &operator*=(const ModInt &p) {
    x = (int) (1LL * x * p.x % mod);
    return *this;
  }

  ModInt &operator/=(const ModInt &p) {
    *this *= p.inverse();
    return *this;
  }

  ModInt operator-() const { return ModInt(-x); }

  ModInt operator+(const ModInt &p) const { return ModInt(*this) += p; }

  ModInt operator-(const ModInt &p) const { return ModInt(*this) -= p; }

  ModInt operator*(const ModInt &p) const { return ModInt(*this) *= p; }

  ModInt operator/(const ModInt &p) const { return ModInt(*this) /= p; }

  bool operator==(const ModInt &p) const { return x == p.x; }

  bool operator!=(const ModInt &p) const { return x != p.x; }

  ModInt inverse() const {
    int a = x, b = mod, u = 1, v = 0, t;
    while(b > 0) {
      t = a / b;
      swap(a -= t * b, b);
      swap(u -= t * v, v);
    }
    return ModInt(u);
  }

  ModInt pow(int64_t n) const {
    ModInt ret(1), mul(x);
    while(n > 0) {
      if(n & 1) ret *= mul;
      mul *= mul;
      n >>= 1;
    }
    return ret;
  }

  friend ostream &operator<<(ostream &os, const ModInt &p) {
    return os << p.x;
  }

  friend istream &operator>>(istream &is, ModInt &a) {
    int64_t t;
    is >> t;
    a = ModInt< mod >(t);
    return (is);
  }

  static int get_mod() { return mod; }
};

template<class _Key, class _Tp, class _Hash, bool DOWNSIZE> class UnorderedMapIterator;
 
template<class _Key, class _Tp, class _Hash = hash<_Key>, bool DOWNSIZE = false>
class UnorderedMap
{
private:
    using iterator = UnorderedMapIterator<_Key, _Tp, _Hash, DOWNSIZE>;
    using value_type = _Tp;
    using data_type = pair<_Key, _Tp>;
    using aligned_pointer = typename aligned_storage<sizeof(value_type), alignof(value_type)>::type;
    friend UnorderedMapIterator<_Key, _Tp, _Hash, DOWNSIZE>;
    struct bucket {
        _Key _key;
        short int _dist;
        bool _last, _end;
        aligned_pointer _value_ptr;
        bucket() noexcept : _dist(-1), _last(false), _end(false){}
        bucket& operator=(const bucket& another) noexcept {
            _key = another._key, _dist = another._dist, _last = another._last, _end = another._end;
            if(!another.empty()){
                new(&_value_ptr) value_type(*reinterpret_cast<const value_type*>(&another._value_ptr));
            }
            return *this;
        }
        ~bucket(){ if(!empty()) _delete(); }
        inline void clear() noexcept { _dist = -1; }
        inline void _delete(){ _dist = -1, value_ptr()->~value_type(); }
        inline bool empty() const noexcept { return (_dist == -1); }
        inline value_type& value() noexcept {
            return *reinterpret_cast<value_type*>(&_value_ptr);
        }
        inline value_type* value_ptr() noexcept {
            return reinterpret_cast<value_type*>(&_value_ptr);
        }
        inline void new_value(value_type&& value){
            new(&_value_ptr) value_type(move(value));
        }
    };
    inline static unsigned int ceilpow2(unsigned int u) noexcept {
        if(u == 0u) return 0u;
        --u, u |= u >> 1, u |= u >> 2, u |= u >> 4, u |= u >> 8;
        return (u | (u >> 16)) + 1u;
    }
    inline static bucket *increment(bucket *cur) noexcept {
        for(++cur; !cur->_end; ++cur){
            if(!cur->empty()) break;
        }
        return cur;
    }
    inline bucket *next_bucket(bucket *cur) const noexcept {
        return cur->_last ? _buckets : cur + 1;
    }
    inline unsigned int make_hash(const _Key& key) const noexcept {
        return _Hash()(key);
    }
    inline float load_rate() const noexcept {
        return (float)_data_count / _bucket_count;
    }
    bucket *insert(bucket *cur, _Key&& key, short int dist, value_type&& value){
        bucket *ret = cur;
        bool flag = false;
        while(true){
            if(cur->empty()){
                cur->_key = move(key), cur->_dist = dist, cur->new_value(move(value));
                if(!flag) ret = cur, flag = true;
                break;
            }else if(dist > cur->_dist){
                swap(key, cur->_key), swap(dist, cur->_dist), swap(value, cur->value());
                if(!flag) ret = cur, flag = true;
            }
            ++dist;
            cur = next_bucket(cur);
        }
        return ret;
    }
    template<class Key>
    bucket *_find(Key&& key, bool push = false){
        unsigned int hash = make_hash(key);
        bucket *cur = _buckets + (hash & _mask);
        short int dist = 0;
        while(dist <= cur->_dist){
            if(key == cur->_key) return cur;
            ++dist, cur = next_bucket(cur);
        }
        if(!push) return _buckets + _bucket_count;
        ++_data_count;
        if(rehash_check()){
            cur = _buckets + (hash & _mask), dist = 0;
        }
        value_type new_value = value_type();
        _Key new_key = forward<Key>(key);
        return insert(cur, move(new_key), dist, move(new_value));
    }
    template<class Data>
    bucket *find_insert(Data&& data){
        const _Key& key = data.first;
        unsigned int hash = make_hash(key);
        bucket *cur = _buckets + (hash & _mask);
        short int dist = 0;
        while(dist <= cur->_dist){
            if(key == cur->_key) return cur;
            ++dist, cur = next_bucket(cur);
        }
        ++_data_count;
        if(rehash_check()){
            cur = _buckets + (hash & _mask), dist = 0;
        }
        data_type new_data = forward<Data>(data);
        return insert(cur, move(new_data.first), dist, move(new_data.second));
    }
    template<typename... Args>
    bucket *emplace(Args&&... args){
        return find_insert(data_type(forward<Args>(args)...));
    }
    bucket *backward_shift(bucket *cur, bool next_ret){
        bucket *next = next_bucket(cur), *ret = cur;
        if(next->_dist < 1) return next_ret ? increment(cur) : cur;
        do {
            cur->_key = next->_key, cur->_dist = next->_dist - 1;
            cur->new_value(move(next->value()));
            cur = next, next = next_bucket(cur);
        }while(next->_dist >= 1);
        cur->clear();
        return ret;
    }
    bucket *erase_impl(bucket *cur, bool next_ret){
        assert(static_cast<size_t>(cur - _buckets) != _bucket_count);
        cur->_delete();
        --_data_count;
        return backward_shift(cur, next_ret);
    }
    bucket *erase_itr(bucket *cur, bool next_ret = true){
        const _Key key = cur->_key;
        return erase_impl(rehash_check() ? _find(key) : cur, next_ret);
    }
    size_t erase_key(const _Key& key){
        rehash_check();
        bucket *cur = _find(key);
        if(static_cast<size_t>(cur - _buckets) == _bucket_count){
            return 0;
        }else{
            erase_impl(_find(key), false);
            return 1;
        }
    }
    bool rehash_check(){
        if(_bucket_count == 0){
            rehash(1u);
            return true;
        }else if(load_rate() >= MAX_LOAD_RATE){
            rehash(_bucket_count * 2u);
            return true;
        }else if(DOWNSIZE){
            if(load_rate() <= MIN_LOAD_RATE && _bucket_count >= DOWNSIZE_THRESHOLD){
                rehash(_bucket_count / 2u);
                return true;
            }
        }
        return false;
    }
    void move_data(bucket *cur){
        insert(_buckets + (make_hash(cur->_key) & _mask), move(cur->_key), 0, move(cur->value()));
    }
    void rehash(unsigned int new_bucket_count){
        UnorderedMap new_unordered_map(new_bucket_count);
        new_unordered_map._data_count = _data_count;
        for(bucket *cur = _buckets; !cur->_end; ++cur){
            if(!cur->empty()){
                new_unordered_map.move_data(cur);
            }
        }
        swap(*this, new_unordered_map);
    }
    friend void swap(UnorderedMap& ump1, UnorderedMap& ump2){
        swap(ump1._bucket_count, ump2._bucket_count);
        swap(ump1._mask, ump2._mask);
        swap(ump1._data_count, ump2._data_count);
        swap(ump1._buckets, ump2._buckets);
    }
 
private:
    unsigned int _bucket_count, _mask, _data_count;
    bucket *_buckets;
public:
    const float MAX_LOAD_RATE = 0.5f;
    const float MIN_LOAD_RATE = 0.1f;
    const unsigned int DOWNSIZE_THRESHOLD = 16u;
    UnorderedMap(unsigned int bucket_size = 0u)
     : _bucket_count(ceilpow2(bucket_size)), _mask(_bucket_count - 1),
        _data_count(0u), _buckets(new bucket[_bucket_count + 1]){
        if(_bucket_count > 0) _buckets[_bucket_count - 1]._last = true;
        else _mask = 0;
        _buckets[_bucket_count]._end = true;
    }
    UnorderedMap(const UnorderedMap& another)
        : _bucket_count(another._bucket_count), _mask(another._mask), _data_count(another._data_count){
        _buckets = new bucket[_bucket_count + 1u];
        for(unsigned int i = 0u; i <= _bucket_count; ++i){
            _buckets[i] = another._buckets[i];
        }
    }
    UnorderedMap(UnorderedMap&& another)
        : _bucket_count(move(another._bucket_count)), _mask(move(another._mask)),
            _data_count(move(another._data_count)), _buckets(another._buckets){
        another._buckets = nullptr;
    }
    UnorderedMap& operator=(const UnorderedMap& another){
        delete[] _buckets;
        _bucket_count = another._bucket_count;
        _mask = another._mask;
        _data_count = another._data_count;
        _buckets = new bucket[_bucket_count + 1u];
        for(unsigned int i = 0u; i <= _bucket_count; ++i){
            _buckets[i] = another._buckets[i];
        }
        return *this;
    }
    UnorderedMap& operator=(UnorderedMap&& another){
        delete[] _buckets;
        _bucket_count = move(another._bucket_count);
        _mask = move(another._mask);
        _data_count = move(another._data_count);
        _buckets = another._buckets;
        another._buckets = nullptr;
        return *this;
    }
    void allocate(unsigned int element_size){
        rehash(ceilpow2(ceil(element_size / MAX_LOAD_RATE) + 1));
    }
    ~UnorderedMap(){ delete[] _buckets; }
    friend ostream& operator<< (ostream& os, UnorderedMap& ump) noexcept {
        for(auto val : ump) os << '{' << val.first << ',' << val.second << "} ";
        return os;
    }
    _Tp& operator[](const _Key& key){ return _find(key, true)->value(); }
    _Tp& operator[](_Key&& key){ return _find(move(key), true)->value(); }
    const _Tp& at(const _Key& key){
        bucket *res = _find(key);
        if(res == _buckets + _bucket_count) __throw_out_of_range("Unordered_Map::at");
        return res->value();
    }
    void clear(){
        UnorderedMap new_unordered_map(0u);
        swap(*this, new_unordered_map);
    }
    size_t size() const noexcept { return _data_count; }
    size_t bucket_count() const noexcept { return _bucket_count; }
    bool empty() const noexcept { return (_data_count == 0); }
    iterator begin() noexcept {
        return (_buckets->empty() && _bucket_count > 0) ? iterator(increment(_buckets)) : iterator(_buckets);
    }
    iterator end() noexcept { return iterator(_buckets + _bucket_count); }
    iterator find(const _Key& key){ return iterator(_find(key)); }
    iterator insert(const data_type& data){ return iterator(find_insert(data)); }
    iterator insert(data_type&& data){ return iterator(find_insert(move(data))); }
    template<typename... Args>
    iterator emplace(Args&&... args){ return iterator(_emplace(forward<Args>(args)...)); }
    size_t erase(const _Key& key){ return erase_key(key); }
    iterator erase(const iterator& itr){ return iterator(erase_itr(itr.bucket_ptr)); }
    void simple_erase(const _Key& key){ erase_key(key); }
    void simple_erase(const iterator& itr){ erase_itr(itr.bucket_ptr, false); }
 
    // DEBUG 用
    short int maximum_distance() const noexcept {
        short int ret = -1;
        for(bucket *cur = _buckets; !cur->_end; ++cur){
            ret = max(ret, cur->_dist);
        }
        return ret;
    }
};
 
template<class _Key, class _Tp, class _Hash, bool DOWNSIZE>
class UnorderedMapIterator {
private:
    friend UnorderedMap<_Key, _Tp, _Hash, DOWNSIZE>;
    typename UnorderedMap<_Key, _Tp, _Hash, DOWNSIZE>::bucket *bucket_ptr;
    using iterator_category = forward_iterator_tag;
    using value_type = pair<const _Key, _Tp>;
    using difference_type = ptrdiff_t;
    using reference = pair<const _Key&, _Tp&>;
 
private:
    UnorderedMapIterator(typename UnorderedMap<_Key, _Tp, _Hash, DOWNSIZE>::bucket *_bucket_ptr)
        noexcept : bucket_ptr(_bucket_ptr){}
public:
    UnorderedMapIterator() noexcept : bucket_ptr(){}
    UnorderedMapIterator(const UnorderedMapIterator& itr) noexcept : bucket_ptr(itr.bucket_ptr){}
    UnorderedMapIterator& operator=(const UnorderedMapIterator& itr)
        & noexcept { return bucket_ptr = itr.bucket_ptr, *this; }
    UnorderedMapIterator& operator=(const UnorderedMapIterator&& itr)
        & noexcept { return bucket_ptr = itr.bucket_ptr, *this; }
    reference operator*() const noexcept { return {bucket_ptr->_key, bucket_ptr->value()}; }
    UnorderedMapIterator& operator++() noexcept {
        return bucket_ptr = UnorderedMap<_Key, _Tp, _Hash, DOWNSIZE>::increment(bucket_ptr), *this;
    }
    UnorderedMapIterator operator++(int) const noexcept {
        return UnorderedMapIterator(UnorderedMap<_Key, _Tp, _Hash, DOWNSIZE>::increment(this->bucket_ptr));
    }
    bool operator==(const UnorderedMapIterator& itr) const noexcept { return !(*this != itr); };
    bool operator!=(const UnorderedMapIterator& itr) const noexcept { return bucket_ptr != itr.bucket_ptr; }
};
// UnorderedMap

using m17 = ModInt<1'000'000'007>;
using m98 = ModInt<998'244'353>;
using MOD = m98;
ll mod = 998'244'353;
// using MOD = m17;
// ll mod = 1'000'000'007;


const int _B = 500500;
V<MOD> g1(_B);
V<MOD> g2(_B);
V<MOD> inverse(_B);
void prepare() {
  g1[0] = g1[1] = g2[0] = g2[1] = 1;
  inverse[0] = 0;
  inverse[1] = 1;
  range(i,2,_B,1) {
    g1[i] = g1[i-1]*i;
    inverse[i] = -inverse[mod%i]*(mod/i);
    g2[i] = g2[i-1]*inverse[i];
  }
}
template<typename T>
MOD cmb(T n, T r) {
  assert(g1[0]==1);
  if (r<0 || r>n) return 0;
  if (g1.size()<=n) {
      int s = g1.size();
      g1.resize(n+1);
      g2.resize(n+1);
      inverse.resize(n+1);
      range(i, s, n+1, 1) {
          g1[i] = g1[i-1]*i;
          inverse[i] = -inverse[mod%i]*(mod/i);
          g2[i] = g2[i-1]*inverse[i];
      }
  }
  r = min(r, n-r);
  return g1[n]*g2[r]*g2[n-r];
}
template<typename T>
MOD perm(T n, T r) {
  if (r<0 || r>n) return 0;
  return g1[n]*g2[n-r];
}


template< typename Mint >
struct NumberTheoreticTransformFriendlyModInt {

  static vector< Mint > dw, idw;
  static int max_base;
  static Mint root;

  NumberTheoreticTransformFriendlyModInt() = default;

  static void init() {
    if(dw.empty()) {
      const unsigned mod = Mint::get_mod();
      assert(mod >= 3 && mod % 2 == 1);
      auto tmp = mod - 1;
      max_base = 0;
      while(tmp % 2 == 0) tmp >>= 1, max_base++;
      root = 2;
      while(root.pow((mod - 1) >> 1) == 1) root += 1;
      assert(root.pow(mod - 1) == 1);
      dw.resize(max_base);
      idw.resize(max_base);
      for(int i = 0; i < max_base; i++) {
        dw[i] = -root.pow((mod - 1) >> (i + 2));
        idw[i] = Mint(1) / dw[i];
      }
    }
  }

  static void ntt(vector< Mint > &a) {
    init();
    const int n = (int) a.size();
    assert((n & (n - 1)) == 0);
    assert(__builtin_ctz(n) <= max_base);
    for(int m = n; m >>= 1;) {
      Mint w = 1;
      for(int s = 0, k = 0; s < n; s += 2 * m) {
        for(int i = s, j = s + m; i < s + m; ++i, ++j) {
          auto x = a[i], y = a[j] * w;
          a[i] = x + y, a[j] = x - y;
        }
        w *= dw[__builtin_ctz(++k)];
      }
    }
  }

  static void intt(vector< Mint > &a, bool f = true) {
    init();
    const int n = (int) a.size();
    assert((n & (n - 1)) == 0);
    assert(__builtin_ctz(n) <= max_base);
    for(int m = 1; m < n; m *= 2) {
      Mint w = 1;
      for(int s = 0, k = 0; s < n; s += 2 * m) {
        for(int i = s, j = s + m; i < s + m; ++i, ++j) {
          auto x = a[i], y = a[j];
          a[i] = x + y, a[j] = (x - y) * w;
        }
        w *= idw[__builtin_ctz(++k)];
      }
    }
    if(f) {
      Mint inv_sz = Mint(1) / n;
      for(int i = 0; i < n; i++) a[i] *= inv_sz;
    }
  }

  static vector< Mint > multiply(vector< Mint > a, vector< Mint > b) {
    int need = a.size() + b.size() - 1;
    int nbase = 1;
    while((1 << nbase) < need) nbase++;
    int sz = 1 << nbase;
    a.resize(sz, 0);
    b.resize(sz, 0);
    ntt(a);
    ntt(b);
    Mint inv_sz = Mint(1) / sz;
    for(int i = 0; i < sz; i++) a[i] *= b[i] * inv_sz;
    intt(a, false);
    a.resize(need);
    return a;
  }
};

template< typename Mint >
vector< Mint > NumberTheoreticTransformFriendlyModInt< Mint >::dw = vector< Mint >();
template< typename Mint >
vector< Mint > NumberTheoreticTransformFriendlyModInt< Mint >::idw = vector< Mint >();
template< typename Mint >
int NumberTheoreticTransformFriendlyModInt< Mint >::max_base = 0;
template< typename Mint >
Mint NumberTheoreticTransformFriendlyModInt< Mint >::root = Mint();


template< typename T >
struct FormalPowerSeriesFriendlyNTT : vector< T > {
  using vector< T >::vector;
  using P = FormalPowerSeriesFriendlyNTT;
  using NTT = NumberTheoreticTransformFriendlyModInt< T >;

  P pre(int deg) const {
    return P(begin(*this), begin(*this) + min((int) this->size(), deg));
  }

  P rev(int deg = -1) const {
    P ret(*this);
    if(deg != -1) ret.resize(deg, T(0));
    reverse(begin(ret), end(ret));
    return ret;
  }

  void shrink() {
    while(this->size() && this->back() == T(0)) this->pop_back();
  }

  P operator+(const P &r) const { return P(*this) += r; }

  P operator+(const T &v) const { return P(*this) += v; }

  P operator-(const P &r) const { return P(*this) -= r; }

  P operator-(const T &v) const { return P(*this) -= v; }

  P operator*(const P &r) const { return P(*this) *= r; }

  P operator*(const T &v) const { return P(*this) *= v; }

  P operator/(const P &r) const { return P(*this) /= r; }

  P operator%(const P &r) const { return P(*this) %= r; }

  P &operator+=(const P &r) {
    if(r.size() > this->size()) this->resize(r.size());
    for(int i = 0; i < r.size(); i++) (*this)[i] += r[i];
    return *this;
  }

  P &operator-=(const P &r) {
    if(r.size() > this->size()) this->resize(r.size());
    for(int i = 0; i < r.size(); i++) (*this)[i] -= r[i];
    return *this;
  }

  // https://judge.yosupo.jp/problem/convolution_mod
  P &operator*=(const P &r) {
    if(this->empty() || r.empty()) {
      this->clear();
      return *this;
    }
    auto ret = NTT::multiply(*this, r);
    return *this = {begin(ret), end(ret)};
  }

  P &operator/=(const P &r) {
    if(this->size() < r.size()) {
      this->clear();
      return *this;
    }
    int n = this->size() - r.size() + 1;
    return *this = (rev().pre(n) * r.rev().inv(n)).pre(n).rev(n);
  }

  P &operator%=(const P &r) {
    *this -= *this / r * r;
    shrink();
    return *this;
  }

  // https://judge.yosupo.jp/problem/division_of_polynomials
  pair< P, P > div_mod(const P &r) {
    P q = *this / r;
    P x = *this - q * r;
    x.shrink();
    return make_pair(q, x);
  }

  P operator-() const {
    P ret(this->size());
    for(int i = 0; i < this->size(); i++) ret[i] = -(*this)[i];
    return ret;
  }

  P &operator+=(const T &r) {
    if(this->empty()) this->resize(1);
    (*this)[0] += r;
    return *this;
  }

  P &operator-=(const T &r) {
    if(this->empty()) this->resize(1);
    (*this)[0] -= r;
    return *this;
  }

  P &operator*=(const T &v) {
    for(int i = 0; i < this->size(); i++) (*this)[i] *= v;
    return *this;
  }

  P dot(P r) const {
    P ret(min(this->size(), r.size()));
    for(int i = 0; i < ret.size(); i++) ret[i] = (*this)[i] * r[i];
    return ret;
  }

  P operator>>(int sz) const {
    if(this->size() <= sz) return {};
    P ret(*this);
    ret.erase(ret.begin(), ret.begin() + sz);
    return ret;
  }

  P operator<<(int sz) const {
    P ret(*this);
    ret.insert(ret.begin(), sz, T(0));
    return ret;
  }

  T operator()(T x) const {
    T r = 0, w = 1;
    for(auto &v : *this) {
      r += w * v;
      w *= x;
    }
    return r;
  }

  P diff() const {
    const int n = (int) this->size();
    P ret(max(0, n - 1));
    for(int i = 1; i < n; i++) ret[i - 1] = (*this)[i] * T(i);
    return ret;
  }

  P integral() const {
    const int n = (int) this->size();
    P ret(n + 1);
    ret[0] = T(0);
    for(int i = 0; i < n; i++) ret[i + 1] = (*this)[i] / T(i + 1);
    return ret;
  }

  // https://judge.yosupo.jp/problem/inv_of_formal_power_series
  // F(0) must not be 0
  P inv(int deg = -1) const {
    assert(((*this)[0]) != T(0));
    const int n = (int) this->size();
    if(deg == -1) deg = n;
    P res(deg);
    res[0] = {T(1) / (*this)[0]};
    for(int d = 1; d < deg; d <<= 1) {
      P f(2 * d), g(2 * d);
      for(int j = 0; j < min(n, 2 * d); j++) f[j] = (*this)[j];
      for(int j = 0; j < d; j++) g[j] = res[j];
      NTT::ntt(f);
      NTT::ntt(g);
      f = f.dot(g);
      NTT::intt(f);
      for(int j = 0; j < d; j++) f[j] = 0;
      NTT::ntt(f);
      for(int j = 0; j < 2 * d; j++) f[j] *= g[j];
      NTT::intt(f);
      for(int j = d; j < min(2 * d, deg); j++) res[j] = -f[j];
    }
    return res;
  }

  // https://judge.yosupo.jp/problem/log_of_formal_power_series
  // F(0) must be 1
  P log(int deg = -1) const {
    assert((*this)[0] == T(1));
    const int n = (int) this->size();
    if(deg == -1) deg = n;
    return (this->diff() * this->inv(deg)).pre(deg - 1).integral();
  }

  // https://judge.yosupo.jp/problem/sqrt_of_formal_power_series
  P sqrt(int deg = -1, const function< T(T) > &get_sqrt = [](T) { return T(1); }) const {
    const int n = (int) this->size();
    if(deg == -1) deg = n;
    if((*this)[0] == T(0)) {
      for(int i = 1; i < n; i++) {
        if((*this)[i] != T(0)) {
          if(i & 1) return {};
          if(deg - i / 2 <= 0) break;
          auto ret = (*this >> i).sqrt(deg - i / 2, get_sqrt);
          if(ret.empty()) return {};
          ret = ret << (i / 2);
          if(ret.size() < deg) ret.resize(deg, T(0));
          return ret;
        }
      }
      return P(deg, 0);
    }
    auto sqr = T(get_sqrt((*this)[0]));
    if(sqr * sqr != (*this)[0]) return {};
    P ret{sqr};
    T inv2 = T(1) / T(2);
    for(int i = 1; i < deg; i <<= 1) {
      ret = (ret + pre(i << 1) * ret.inv(i << 1)) * inv2;
    }
    return ret.pre(deg);
  }

  P sqrt(const function< T(T) > &get_sqrt, int deg = -1) const {
    return sqrt(deg, get_sqrt);
  }

  // https://judge.yosupo.jp/problem/exp_of_formal_power_series
  // F(0) must be 0
  P exp(int deg = -1) const {
    if(deg == -1) deg = this->size();
    assert((*this)[0] == T(0));

    P inv;
    inv.reserve(deg + 1);
    inv.push_back(T(0));
    inv.push_back(T(1));

    auto inplace_integral = [&](P &F) -> void {
      const int n = (int) F.size();
      auto mod = T::get_mod();
      while((int) inv.size() <= n) {
        int i = inv.size();
        inv.push_back((-inv[mod % i]) * (mod / i));
      }
      F.insert(begin(F), T(0));
      for(int i = 1; i <= n; i++) F[i] *= inv[i];
    };

    auto inplace_diff = [](P &F) -> void {
      if(F.empty()) return;
      F.erase(begin(F));
      T coeff = 1, one = 1;
      for(int i = 0; i < (int) F.size(); i++) {
        F[i] *= coeff;
        coeff += one;
      }
    };

    P b{1, 1 < (int) this->size() ? (*this)[1] : 0}, c{1}, z1, z2{1, 1};
    for(int m = 2; m < deg; m *= 2) {
      auto y = b;
      y.resize(2 * m);
      NTT::ntt(y);
      z1 = z2;
      P z(m);
      for(int i = 0; i < m; ++i) z[i] = y[i] * z1[i];
      NTT::intt(z);
      fill(begin(z), begin(z) + m / 2, T(0));
      NTT::ntt(z);
      for(int i = 0; i < m; ++i) z[i] *= -z1[i];
      NTT::intt(z);
      c.insert(end(c), begin(z) + m / 2, end(z));
      z2 = c;
      z2.resize(2 * m);
      NTT::ntt(z2);
      P x(begin(*this), begin(*this) + min< int >(this->size(), m));
      inplace_diff(x);
      x.push_back(T(0));
      NTT::ntt(x);
      for(int i = 0; i < m; ++i) x[i] *= y[i];
      NTT::intt(x);
      x -= b.diff();
      x.resize(2 * m);
      for(int i = 0; i < m - 1; ++i) x[m + i] = x[i], x[i] = T(0);
      NTT::ntt(x);
      for(int i = 0; i < 2 * m; ++i) x[i] *= z2[i];
      NTT::intt(x);
      x.pop_back();
      inplace_integral(x);
      for(int i = m; i < min< int >(this->size(), 2 * m); ++i) x[i] += (*this)[i];
      fill(begin(x), begin(x) + m, T(0));
      NTT::ntt(x);
      for(int i = 0; i < 2 * m; ++i) x[i] *= y[i];
      NTT::intt(x);
      b.insert(end(b), begin(x) + m, end(x));
    }
    return P{begin(b), begin(b) + deg};
  }

  // https://judge.yosupo.jp/problem/pow_of_formal_power_series
  P pow(int64_t k, int deg = -1) const {
    const int n = (int) this->size();
    if(deg == -1) deg = n;
    for(int i = 0; i < n; i++) {
      if((*this)[i] != T(0)) {
        T rev = T(1) / (*this)[i];
        P ret = (((*this * rev) >> i).log() * k).exp() * ((*this)[i].pow(k));
        if(i * k > deg) return P(deg, T(0));
        ret = (ret << (i * k)).pre(deg);
        if(ret.size() < deg) ret.resize(deg, T(0));
        return ret;
      }
    }
    return *this;
  }

  P mod_pow(int64_t k, P g) const {
    P modinv = g.rev().inv();
    auto get_div = [&](P base) {
      if(base.size() < g.size()) {
        base.clear();
        return base;
      }
      int n = base.size() - g.size() + 1;
      return (base.rev().pre(n) * modinv.pre(n)).pre(n).rev(n);
    };
    P x(*this), ret{1};
    while(k > 0) {
      if(k & 1) {
        ret *= x;
        ret -= get_div(ret) * g;
        ret.shrink();
      }
      x *= x;
      x -= get_div(x) * g;
      x.shrink();
      k >>= 1;
    }
    return ret;
  }

  // https://judge.yosupo.jp/problem/polynomial_taylor_shift
  P taylor_shift(T c) const {
    int n = (int) this->size();
    vector< T > fact(n), rfact(n);
    fact[0] = rfact[0] = T(1);
    for(int i = 1; i < n; i++) fact[i] = fact[i - 1] * T(i);
    rfact[n - 1] = T(1) / fact[n - 1];
    for(int i = n - 1; i > 1; i--) rfact[i - 1] = rfact[i] * T(i);
    P p(*this);
    for(int i = 0; i < n; i++) p[i] *= fact[i];
    p = p.rev();
    P bs(n, T(1));
    for(int i = 1; i < n; i++) bs[i] = bs[i - 1] * c * rfact[i] * fact[i - 1];
    p = (p * bs).pre(n);
    p = p.rev();
    for(int i = 0; i < n; i++) p[i] *= rfact[i];
    return p;
  }
};

using FPS = FormalPowerSeriesFriendlyNTT< MOD >;

// usage: 
// FPS f, g;
// f.reserve(s+1);
// g.reserve(s+1);
// rep(j,s+1) {
//     f.emplace_back(dpa[i][j]);
//     g.emplace_back(dpb[n-i][j]);
// }
// auto h = f*g;

// multiply functions


void Main(){
    ll n;
    cin >> n;
    // n = 100000;
    V<ll> a(n);
    ll t = 0;
    rep(i,n) {
        cin >> a[i];
        // a[i] = 10000;
        t += a[i];
    }
    MOD ans = MOD(2).pow(n-1) * t;
    ll M = 999630629;
    if (t>=M) {
        ll m = t - M;
        FPS f(1);
        V<int> count(10001);
        for (auto v : a) count[v]++;
        rep(v,10001) {
            if (count[v]) {
                FPS g(v+1);
                g[0] = g[v] = 1;
                f += g.log(v) * v;
            }
        }
        auto h = f.exp();
        MOD val = 0;
        rep(i,m+1) val += h[i];
        ans -= MOD(M) * val;
    }
    cout << ans << "\n";
}

int main(void){
    std::ifstream in("tmp_in");
    if (TEST) {
        std::cin.rdbuf(in.rdbuf());
        std::cout << std::fixed << std::setprecision(15);
    } else {
        std::cin.tie(nullptr);
        std::ios_base::sync_with_stdio(false);
        std::cout << std::fixed << std::setprecision(15);
    }
    Main();
}