#include <iostream>
#include <utility>
#include <vector>
#include <list>
#include <unordered_set>
#include <set>
#include <unordered_map>
#include <map>
#include <stack>
#include <queue>
#include <cassert>
#include <cmath>
#include <algorithm>
#include <cstdlib>
#include <cstdio>
#include <fstream>
#include <tuple>
#include <type_traits>

//....................................................................................................................//

using std::cin;
using std::cout;
using std::vector;
using std::list;
using std::unordered_set;
using std::set;
using std::map;
using std::unordered_map;
using std::multimap;
using std::unordered_multimap;
using std::pair;
using std::make_pair;
using std::min;
using std::max;
using std::swap;
using std::stack;
using std::deque;
using std::queue;
using std::ios_base;
using std::ifstream;
using std::ofstream;
using std::string;
using std::tuple;
using std::make_tuple;
using std::get;
using std::upper_bound;
using std::lower_bound;
using std::max_element;
using std::min_element;
using std::multiset;
using std::greater;
using std::priority_queue;
using std::less;
using std::endl;
//....................................................................................................................//

typedef long long ll;
typedef unsigned long long ull;

//[[maybe_unused]] const string endl = "\n";
[[maybe_unused]] const string sep = " ";
[[maybe_unused]] const string yes = "YES\n";
[[maybe_unused]] const string no = "NO\n";
[[maybe_unused]] const string ng = "NG\n";
[[maybe_unused]] const string ok = "OK\n";
[[maybe_unused]] const double pi = acos(-1);
[[maybe_unused]] const ull modular = 1000000007;

//....................................................................................................................//



class BigInteger {
public:
    BigInteger();

    BigInteger(int int_to_big_integer);

    BigInteger(const std::string &string_to_big_integer);

    friend BigInteger operator+(const BigInteger &big_integer_a,
                                const BigInteger &big_integer_b);

    friend BigInteger operator-(const BigInteger &big_integer_a,
                                const BigInteger &big_integer_b);

    friend BigInteger operator*(const BigInteger &big_integer_a,
                                const BigInteger &big_integer_b);

    friend BigInteger operator/(const BigInteger &big_integer_a,
                                const BigInteger &big_integer_b);

    friend BigInteger operator%(const BigInteger &big_integer_a,
                                const BigInteger &big_integer_b);

    BigInteger &operator+=(const BigInteger &big_integer_a);

    BigInteger &operator-=(const BigInteger &big_integer_a);

    BigInteger &operator*=(const BigInteger &big_integer_a);

    BigInteger &operator%=(const BigInteger &big_integer_a);

    BigInteger &operator/=(const BigInteger &big_integer_a);

    bool operator<(BigInteger const &big_integer_a) const;

    bool operator>(BigInteger const &big_integer_a) const;

    bool operator<=(BigInteger const &big_integer_a) const;

    bool operator>=(BigInteger const &big_integer_a) const;

    bool operator==(BigInteger const &big_integer_a) const;

    bool operator!=(BigInteger const &big_integer_a) const;

    BigInteger &operator++();

    const BigInteger operator++(int);

    BigInteger &operator--();

    const BigInteger operator--(int);

    BigInteger operator-();

    std::string toString() const;

    explicit operator bool();

    friend std::istream &operator>>(std::istream &in, BigInteger &big_integer_a);

    friend std::ostream &operator<<(std::ostream &out,
                                    const BigInteger &big_integer_a);

private:
    bool is_positive_ = true;

    std::vector<int> digits_;

    void EraseLeadingZeros();

    bool IsZero() const;

    static void SubIfPositive(BigInteger &big_integer_a,
                              const BigInteger &big_integer_b);

    static void Sum(BigInteger &big_integer_a, const BigInteger &big_integer_b);

    BigInteger Abs() const;

    friend std::vector<int> NaiveMultiplication(const std::vector<int> &digits_a,
                                                const std::vector<int> &digits_b);

    friend void Normalize(std::vector<int> &digits);

    friend std::vector<int> FastMultiplication(const std::vector<int> &digits_a,
                                               const std::vector<int> &digits_b);

    bool DifferentSigns(const BigInteger &big_integer_a) const;
};

//...........................................................................................//

std::vector<int> NaiveMultiplication(const std::vector<int> &digits_a,
                                     const std::vector<int> &digits_b) {
    auto size = digits_a.size();
    std::vector<int> digits_result(2 * size);
    for (size_t i = 0; i < size; ++i) {
        for (size_t j = 0; j < size; ++j) {
            digits_result[i + j] += digits_a[i] * digits_b[j];
        }
    }
    return digits_result;
}

void Normalize(std::vector<int> &digits) {
    for (size_t i = 0; i < digits.size(); ++i) {
        digits[i + 1] += digits[i] / 10;
        digits[i] %= 10;
    }
}

std::vector<int> FastMultiplication(const std::vector<int> &digits_a,
                                    const std::vector<int> &digits_b) {
    auto size = digits_a.size();
    std::vector<int> digits_result(2 * size);
//    if (size <= 1000) {
    return NaiveMultiplication(digits_a, digits_b);
//    }
//    auto half_of_size = size / 2;
//    std::vector<int> right_digits_a{digits_a.begin(),
//                                    digits_a.begin() + half_of_size};
//    std::vector<int> left_digits_a{digits_a.begin() + half_of_size,
//                                   digits_a.end()};
//    std::vector<int> right_digits_b{digits_b.begin(),
//                                    digits_b.begin() + half_of_size};
//    std::vector<int> left_digits_b{digits_b.begin() + half_of_size,
//                                   digits_b.end()};
//    std::vector<int> left_left = FastMultiplication(left_digits_a, left_digits_b);
//    std::vector<int> right_right =
//            FastMultiplication(right_digits_a, right_digits_b);
//    std::vector<int> sum_left_right_a(half_of_size);
//    std::vector<int> sum_left_right_b(half_of_size);
//    for (size_t i = 0; i < half_of_size; ++i) {
//        sum_left_right_a[i] = left_digits_a[i] + right_digits_a[i];
//        sum_left_right_b[i] = left_digits_b[i] + right_digits_b[i];
//    }
//    std::vector<int> sum_sum =
//            FastMultiplication(sum_left_right_a, sum_left_right_b);
//    for (size_t i = 0; i < size; ++i) {
//        sum_sum[i] -= right_right[i] + left_left[i];
//        digits_result[i] = right_right[i];
//    }
//    for (size_t i = size; i < 2 * size; ++i) {
//        digits_result[i] = left_left[i - size];
//    }
//    for (size_t i = half_of_size; i < size + half_of_size; ++i) {
//        digits_result[i] += sum_sum[i - half_of_size];
//    }
//    Normalize(digits_result);
//    return digits_result;
}

BigInteger::BigInteger() : is_positive_(true) {}

BigInteger::BigInteger(int int_to_big_integer) {
    if (int_to_big_integer < 0) {
        is_positive_ = false;
        int_to_big_integer *= -1;
    } else if (int_to_big_integer == 0) {
        digits_.push_back(0);
        is_positive_ = true;
    }
    while (int_to_big_integer != 0) {
        digits_.push_back(int_to_big_integer % 10);
        int_to_big_integer /= 10;
    }
}

BigInteger::BigInteger(const std::string &string_to_big_integer) {
    is_positive_ = true;
    if (!string_to_big_integer.empty()) {
        for (size_t i = string_to_big_integer.length() - 1; i > 0; --i)
            digits_.push_back(string_to_big_integer[i] - '0');
        if (string_to_big_integer[0] == '-')
            is_positive_ = false;
        else
            digits_.push_back(string_to_big_integer[0] - '0');
    }
}

//...........................................................................................//

BigInteger operator+(const BigInteger &big_integer_a,
                     const BigInteger &big_integer_b) {
    BigInteger big_integer_sum = big_integer_a;
    return big_integer_sum += big_integer_b;
}

BigInteger operator-(const BigInteger &big_integer_a,
                     const BigInteger &big_integer_b) {
    BigInteger big_integer_sub = big_integer_a;
    return big_integer_sub -= big_integer_b;
}

BigInteger operator*(const BigInteger &big_integer_a,
                     const BigInteger &big_integer_b) {
    BigInteger big_integer_multiplication = big_integer_a;
    return big_integer_multiplication *= big_integer_b;
}

BigInteger operator/(const BigInteger &big_integer_a,
                     const BigInteger &big_integer_b) {
    BigInteger big_integer_division = big_integer_a;
    return big_integer_division /= big_integer_b;
}

BigInteger operator%(const BigInteger &big_integer_a,
                     const BigInteger &big_integer_b) {
    BigInteger big_integer_division_with_remainder = big_integer_a;
    return big_integer_division_with_remainder %= big_integer_b;
}

//...........................................................................................//

BigInteger &BigInteger::operator+=(const BigInteger &big_integer_a) {
    if (!DifferentSigns(big_integer_a)) {
        Sum(*this, big_integer_a);
        return *this;
    } else {
        if (this->Abs() >= big_integer_a.Abs()) {
            SubIfPositive(*this, big_integer_a);
            return *this;
        } else {
            BigInteger big_integer_temp_this = *this;
            BigInteger big_integer_temp_a = big_integer_a;
            SubIfPositive(big_integer_temp_a, big_integer_temp_this);
            return *this = big_integer_temp_a;
        }
    }
}

BigInteger &BigInteger::operator-=(const BigInteger &big_integer_a) {
    if (!DifferentSigns(big_integer_a)) {
        if (this->Abs() >= big_integer_a.Abs()) {
            SubIfPositive(*this, big_integer_a);
            return *this;
        } else {
            BigInteger big_integer_temp_this = *this;
            BigInteger big_integer_temp_a = big_integer_a;
            SubIfPositive(big_integer_temp_a, big_integer_temp_this);
            big_integer_temp_a.is_positive_ = !big_integer_temp_a.is_positive_;
            return *this = big_integer_temp_a;
        }
    } else {
        Sum(*this, big_integer_a);
        return *this;
    }
}

BigInteger &BigInteger::operator*=(const BigInteger &big_integer_a) {
    size_t size = std::max(big_integer_a.digits_.size(), digits_.size());
    BigInteger big_integer_result;
    auto big_integer_temp_a = big_integer_a;
    big_integer_temp_a.digits_.resize(size);
    digits_.resize(size);
    digits_ = FastMultiplication(big_integer_temp_a.digits_, digits_);
    Normalize(digits_);
    EraseLeadingZeros();
    is_positive_ = !(DifferentSigns(big_integer_a));
    return *this;
}

BigInteger &BigInteger::operator/=(const BigInteger &big_integer_a) {
    BigInteger big_integer_result;
    big_integer_result.digits_.resize(digits_.size());
    BigInteger big_integer_current;
    for (int i = digits_.size() - 1; i >= 0; --i) {
        big_integer_current.digits_.insert(big_integer_current.digits_.begin(), 0);
        big_integer_current.EraseLeadingZeros();
        big_integer_current.digits_[0] = digits_[i];
        size_t new_digit = 0;
        size_t left = 0, right = 9;
        while (left <= right) {
            size_t middle = (left + right) / 2;
            if ((big_integer_a.Abs() * middle).Abs() <= big_integer_current.Abs()) {
                new_digit = middle;
                left = middle + 1;
            } else
                right = middle - 1;
        }
        big_integer_current -= big_integer_a.Abs() * new_digit;
        big_integer_result.digits_[i] = new_digit;
    }

    big_integer_result.is_positive_ = !(DifferentSigns(big_integer_a));
    big_integer_result.EraseLeadingZeros();
    *this = big_integer_result;
    return *this;
}

BigInteger &BigInteger::operator%=(const BigInteger &big_integer_a) {
    BigInteger big_integer_result;
    big_integer_result.digits_.resize(digits_.size());
    BigInteger big_integer_current;
    for (int i = digits_.size() - 1; i >= 0; --i) {
        big_integer_current.digits_.insert(big_integer_current.digits_.begin(), 0);
        big_integer_current.EraseLeadingZeros();
        big_integer_current.digits_[0] = digits_[i];
        size_t new_digit = 0;
        size_t left = 0, right = 9;
        while (left <= right) {
            size_t middle = (left + right) / 2;
            if ((big_integer_a.Abs() * middle).Abs() <= big_integer_current.Abs()) {
                new_digit = middle;
                left = middle + 1;
            } else
                right = middle - 1;
        }
        big_integer_result.digits_[i] = new_digit;
        big_integer_current = big_integer_current - big_integer_a.Abs() * new_digit;
    }
    big_integer_current.EraseLeadingZeros();
    big_integer_current.is_positive_ = is_positive_;
    if (big_integer_current.digits_.empty())
        big_integer_current.digits_.push_back(0);
    *this = big_integer_current;
    return *this;
}

//...........................................................................................//

bool BigInteger::operator<(BigInteger const &big_integer_a) const {
    if (IsZero() && big_integer_a.IsZero()) {
        return false;
    }
    if (DifferentSigns(big_integer_a)) {
        return !is_positive_;
    }
    if (digits_.size() > big_integer_a.digits_.size()) {
        return !is_positive_;
    }
    if (digits_.size() < big_integer_a.digits_.size()) {
        return is_positive_;
    }
    for (int i = digits_.size() - 1; i >= 0; --i) {
        if (digits_[i] > big_integer_a.digits_[i]) {
            return !is_positive_;
        }
        if (digits_[i] < big_integer_a.digits_[i]) {
            return is_positive_;
        }
    }
    return false;
}

bool BigInteger::operator>(BigInteger const &big_integer_a) const {
    return (big_integer_a < *this);
}

bool BigInteger::operator<=(BigInteger const &big_integer_a) const {
    return !(*this > big_integer_a);
}

bool BigInteger::operator>=(BigInteger const &big_integer_a) const {
    return !(*this < big_integer_a);
}

bool BigInteger::operator==(BigInteger const &big_integer_a) const {
    return (*this >= big_integer_a) && (*this <= big_integer_a);
}

bool BigInteger::operator!=(BigInteger const &big_integer_a) const {
    return !(*this == big_integer_a);
}

//...........................................................................................//

BigInteger &BigInteger::operator++() {
    *this += 1;
    return *this;
}

const BigInteger BigInteger::operator++(int) {
    BigInteger big_integer_temp_this = *this;
    ++(*this);
    return big_integer_temp_this;
}

BigInteger &BigInteger::operator--() {
    *this -= 1;
    return *this;
}

const BigInteger BigInteger::operator--(int) {
    BigInteger big_integer_temp_this = *this;
    --(*this);
    return big_integer_temp_this;
}

//...........................................................................................//

std::istream &operator>>(std::istream &in, BigInteger &big_integer_a) {
    std::string string_to_big_integer;
    in >> string_to_big_integer;
    big_integer_a = BigInteger(string_to_big_integer);
    return in;
}

std::ostream &operator<<(std::ostream &out, const BigInteger &big_integer_a) {
    out << big_integer_a.toString();
    return out;
}

BigInteger BigInteger::operator-() {
    BigInteger big_integer_temp_this = *this;
    big_integer_temp_this.is_positive_ = !this->is_positive_;
    if (IsZero()) {
        big_integer_temp_this.is_positive_ = true;
        big_integer_temp_this.digits_[0] = 0;
        return big_integer_temp_this;
    } else
        return big_integer_temp_this;
}

//...........................................................................................//

std::string BigInteger::toString() const {
    std::string s = is_positive_ ? "" : "-";
    if (IsZero()) {
        s = '0';
        return s;
    }
    bool leading_zero = true;
    for (int i = digits_.size() - 1; i >= 0; --i) {
        if (!(digits_[i] == 0 && leading_zero)) {
            leading_zero = false;
            s += digits_[i] + '0';
        }
    }
    if (s.empty() || (s[0] == '-' && s.length() == 1)) s = '0';
    return s;
}

//...........................................................................................//

BigInteger::operator bool() { return !this->IsZero(); }

//...........................................................................................//

BigInteger BigInteger::Abs() const {
    BigInteger big_integer_temp_this = (*this);
    big_integer_temp_this.is_positive_ = true;
    return big_integer_temp_this;
}

void BigInteger::EraseLeadingZeros() {
    while (!digits_.empty() && *digits_.rbegin() == 0) {
        digits_.pop_back();
    }
    if (digits_.empty()) {
        is_positive_ = true;
        digits_.push_back(0);
    }
}

void BigInteger::SubIfPositive(BigInteger &big_integer_a,
                               const BigInteger &big_integer_b) {
    int carry = 0;
    for (size_t i = 0; i < big_integer_b.digits_.size() || carry; ++i) {
        if (i < big_integer_b.digits_.size()) {
            big_integer_a.digits_[i] -= carry + big_integer_b.digits_[i];
        } else {
            big_integer_a.digits_[i] -= carry;
        }
        carry = (big_integer_a.digits_[i] < 0);
        if (carry != 0) big_integer_a.digits_[i] += 10;
    }
    big_integer_a.EraseLeadingZeros();
}

void BigInteger::Sum(BigInteger &big_integer_a,
                     const BigInteger &big_integer_b) {
    int carry = 0;
    for (size_t i = 0; i < big_integer_a.digits_.size() ||
                       i < big_integer_b.digits_.size() || carry;
         ++i) {
        if (i == big_integer_a.digits_.size()) {
            big_integer_a.digits_.push_back(0);
        }
        if (i < big_integer_b.digits_.size()) {
            big_integer_a.digits_[i] += carry + big_integer_b.digits_[i];
        } else {
            big_integer_a.digits_[i] += carry;
        }
        carry = big_integer_a.digits_[i] / 10;
        big_integer_a.digits_[i] %= 10;
    }
    big_integer_a.EraseLeadingZeros();
}

bool BigInteger::IsZero() const {
    return digits_[0] == 0 && digits_.size() == 1;
}

bool BigInteger::DifferentSigns(const BigInteger &big_integer_a) const {
    return is_positive_ ^ big_integer_a.is_positive_;
}

//...........................................................................................//


template<class T>
T min(const T &a, const T &b, const T &c) {
    return min(a, min(b, c));
}

template<class T>
T max(const T &a, const T &b, const T &c) {
    return max(a, max(b, c));
}

typedef vector<vector<ll>> graph;

vector<ll> lca_h, lca_dfs_list, lca_first, lca_tree;
vector<char> lca_dfs_used;

void lca_dfs(const graph &g, ll v, ll h = 1) {
    lca_dfs_used[v] = true;
    lca_h[v] = h;
    lca_dfs_list.push_back(v);
    for (auto i = g[v].begin(); i != g[v].end(); ++i)
        if (!lca_dfs_used[*i]) {
            lca_dfs(g, *i, h + 1);
            lca_dfs_list.push_back(v);
        }
}

void lca_build_tree(ll i, ll l, ll r) {
    if (l == r)
        lca_tree[i] = lca_dfs_list[l];
    else {
        ll m = (l + r) >> 1;
        lca_build_tree(i + i, l, m);
        lca_build_tree(i + i + 1, m + 1, r);
        if (lca_h[lca_tree[i + i]] < lca_h[lca_tree[i + i + 1]])
            lca_tree[i] = lca_tree[i + i];
        else
            lca_tree[i] = lca_tree[i + i + 1];
    }
}

void lca_prepare(const graph &g, ll root) {
    ll n = (ll) g.size();
    lca_h.resize(n);
    lca_dfs_list.reserve(n * 2);
    lca_dfs_used.assign(n, 0);

    lca_dfs(g, root);

    ll m = (ll) lca_dfs_list.size();
    lca_tree.assign(lca_dfs_list.size() * 4 + 1, -1);
    lca_build_tree(1, 0, m - 1);

    lca_first.assign(n, -1);
    for (ll i = 0; i < m; ++i) {
        ll v = lca_dfs_list[i];
        if (lca_first[v] == -1)
            lca_first[v] = i;
    }
}

ll lca_tree_min(ll i, ll sl, ll sr, ll l, ll r) {
    if (sl == l && sr == r)
        return lca_tree[i];
    ll sm = (sl + sr) >> 1;
    if (r <= sm)
        return lca_tree_min(i + i, sl, sm, l, r);
    if (l > sm)
        return lca_tree_min(i + i + 1, sm + 1, sr, l, r);
    ll ans1 = lca_tree_min(i + i, sl, sm, l, sm);
    ll ans2 = lca_tree_min(i + i + 1, sm + 1, sr, sm + 1, r);
    return lca_h[ans1] < lca_h[ans2] ? ans1 : ans2;
}

ll lca(ll a, ll b) {
    ll left = lca_first[a],
            right = lca_first[b];
    if (left > right) swap(left, right);
    return lca_tree_min(1, 0, (ll) lca_dfs_list.size() - 1, left, right);
}


ll gcd(ll a, ll b) {
    if (a == 0) {
        return b;
    }
    return gcd(b % a, a);
}

ll fast_power(ll a, ll pow) {
    if (pow == 0) {
        return 1;
    }
    if (pow % 2 == 0) {
        auto x = fast_power(a, pow / 2);
        return (x * x);
    } else {
        auto x = fast_power(a, pow - 1);
        return (a * x);
    }
}

ll fast_power(ll a, ll pow, ll mod) {
    if (pow == 0) {
        return 1;
    }
    if (pow % 2 == 0) {
        auto x = fast_power(a, pow / 2, mod);
        return (x * x) % mod;
    } else {
        auto x = fast_power(a, pow - 1, mod);
        return (a * x) % mod;
    }

}

void getFactorialsModular(const ll n, const ll mod, vector<ll> &F) {
    F.resize(n + 1);
    F[0] = 1;
    for (ll i = 1; i <= n; ++i) {
        F[i] = (F[i - 1] * i) % mod;
    }
}

void getFactorialsAndInverseModular(const ll n, const ll mod, vector<ll> &F, vector<ll> &I) {
    getFactorialsModular(n, mod, F);
    I.resize(n + 1);
    I[0] = 1;
    for (ll i = 1; i <= n; ++i) {
        I[i] = fast_power(F[i], mod - 2, mod);
    }
}

ll CnkMod(ll n, ll k, ll mod, vector<ll> &F, vector<ll> &I) {
    return (((F[n] * I[k]) % mod) * I[n - k]) % mod;
}

ll CnkWithRepetitionMod(ll n, ll k, ll mod, vector<ll> &F, vector<ll> &I) {
    return CnkMod(n + k - 1, k - 1, mod, F, I);
}

/*
MIT License
Copyright (c) 2019 ouuan
Permission is hereby granted, free of charge, to any person obtaining a copy
of this software and associated documentation files (the "Software"), to deal
in the Software without restriction, including without limitation the rights
to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
copies of the Software, and to permit persons to whom the Software is
furnished to do so, subject to the following conditions:
The above copyright notice and this permission notice shall be included in all
copies or substantial portions of the Software.
THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
SOFTWARE.
*/

// for further information and bug tracking,
// please visit https://github.com/ouuan/segmentTree


template<typename valueType, typename tagType>
class segmentTreeNode {
public:
    int id, left, right;
    valueType val;
    tagType tag;
};

template<typename valueType,
        typename tagType,
        valueType (*merge)(valueType, valueType),
        void (*update)(segmentTreeNode<valueType, tagType> &, tagType)>
class segmentTree {
private:
    std::vector<segmentTreeNode<valueType, tagType> > nodes;
    int leftRange, rightRange;
    valueType valueZero;
    tagType tagZero;

    void push_up(int cur) {
        nodes[cur].val = merge(nodes[cur << 1].val, nodes[cur << 1 | 1].val);
    }

    void push_down(int cur) {
        update(nodes[cur << 1], nodes[cur].tag);
        update(nodes[cur << 1 | 1], nodes[cur].tag);
        nodes[cur].tag = tagZero;
    }

    void build(int cur, int l, int r, const std::vector<valueType> &initValue) {
        nodes[cur].id = cur;
        nodes[cur].left = l;
        nodes[cur].right = r;
        nodes[cur].tag = tagZero;
        if (l == r - 1) nodes[cur].val = initValue[l - leftRange];
        else {
            build(cur << 1, l, (l + r) >> 1, initValue);
            build(cur << 1 | 1, (l + r) >> 1, r, initValue);
            push_up(cur);
        }
    }

    void init(const std::vector<valueType> &_initValue,
              const valueType &_valueZero,
              const tagType &_tagZero) {
        valueZero = _valueZero;
        tagZero = _tagZero;
        nodes.resize((rightRange - leftRange) << 2);
        build(1, leftRange, rightRange, _initValue);
    }

    void modify(int cur, int l, int r, int L, int R, const tagType &tag) {
        if (l >= R || r <= L) return;
        if (L <= l && r <= R) update(nodes[cur], tag);
        else {
            push_down(cur);
            modify(cur << 1, l, (l + r) >> 1, L, R, tag);
            modify(cur << 1 | 1, (l + r) >> 1, r, L, R, tag);
            push_up(cur);
        }
    }

    valueType query(int cur, int l, int r, int L, int R) {
        if (l >= R || r <= L) return valueZero;
        if (L <= l && r <= R) return nodes[cur].val;
        push_down(cur);
        return merge(query(cur << 1, l, (l + r) >> 1, L, R),
                     query(cur << 1 | 1, (l + r) >> 1, r, L, R));
    }

public:
    segmentTree() {}

    segmentTree(int _leftRange,
                int _rightRange,
                const std::vector<valueType> &_initValue,
                const valueType &_valueZero,
                const tagType &_tagZero) {
        leftRange = _leftRange;
        rightRange = _rightRange;
        init(_initValue, _valueZero, _tagZero);
    }

    segmentTree(int size,
                const std::vector<valueType> &_initValue,
                const valueType &_valueZero,
                const tagType &_tagZero) {
        leftRange = 1;
        rightRange = size + 1;
        init(_initValue, _valueZero, _tagZero);
    }

    void modify(int l, int r, const tagType &tag) {
        modify(1, leftRange, rightRange, l, r, tag);
    }

    void modify(int p, const tagType &tag) {
        modify(p, p + 1, tag);
    }

    valueType query(int l, int r) {
        return query(1, leftRange, rightRange, l, r);
    }

    valueType query(int p) {
        return query(p, p + 1);
    }
};

ll merge(ll x, ll y) {
    return x + y;
}

void update(segmentTreeNode<ll, ll> &node, ll x) {
    node.val += x;
    node.val %= 2;
}


int main() {
    cin.tie(nullptr);
    cout.tie(nullptr);
    std::ios_base::sync_with_stdio(false);
    cout.precision(15);
    ll h, w, k;
    cin >> h >> w >> k;
    vector<string> G(h);
    for (int i = 0; i < h; ++i) {
        cin >> G[i];
    }
    ll counter = 0;
    for (ll i = 0; i < (1LL << h); ++i) {
        set<ll> Rows;
        ll ti = i;
        ll c = 0;
        while (ti) {
            if (ti % 2) {
                Rows.insert(c);
            }
            ++c;
            ti /= 2;
        }
        for (ll j = 0; j < (1LL << w); ++j) {
            set<ll> Cols;
            ll tj = j;
            ll c = 0;
            while (tj) {
                if (tj % 2) {
                    Cols.insert(c);
                }
                ++c;
                tj /= 2;
            }
            ll blacks = 0;
            for (int a = 0; a < h; ++a) {
                for (int b = 0; b < w; ++b) {
                    if (G[a][b] == '#' && !Rows.count(a) && !Cols.count(b)) {
                        ++blacks;
                    }
                }
            }
            if (blacks == k) {
                ++counter;
            }
        }
    }
    cout << counter << endl;
}


Measured memory_bytes: 3668