p480.ans1
======================
1
1 0 8

=================
p480.cpp
======================
#include <bits/stdc++.h>

using namespace std;

template<typename T1, typename T2>
ostream& operator<<(ostream& out, const pair<T1, T2>& x) {
    return out << x.first << ' ' << x.second;
}

template<typename T1, typename T2>
istream& operator>>(istream& in, pair<T1, T2>& x) {
    return in >> x.first >> x.second;
}

template<typename T>
istream& operator>>(istream& in, vector<T>& a) {
    for(auto& x: a) {
        in >> x;
    }
    return in;
};

template<typename T>
ostream& operator<<(ostream& out, const vector<T>& a) {
    for(auto x: a) {
        out << x << ' ';
    }
    return out;
};

int c, k;
vector<vector<int>> a;

void read() {
    cin >> c >> k;
    a.assign(c, vector<int>(k));
    for(int i = 0; i < c; i++) {
        for(int j = 0; j < k; j++) {
            cin >> a[i][j];
        }
    }
}

vector<int> normalize_base8(const vector<int>& coeffs) {
    vector<int> digits;
    digits.reserve(coeffs.size() + 4);
    int64_t carry = 0;
    for(int v: coeffs) {
        int64_t cur = (int64_t)v + carry;
        digits.push_back((int)(cur % 8));
        carry = cur / 8;
    }

    while(carry > 0) {
        digits.push_back((int)(carry % 8));
        carry /= 8;
    }

    while(!digits.empty() && digits.back() == 0) {
        digits.pop_back();
    }

    return digits;
}

int cmp_digits(const vector<int>& x, const vector<int>& y) {
    if(x.size() != y.size()) {
        return x.size() < y.size() ? -1 : 1;
    }

    for(int i = (int)x.size() - 1; i >= 0; i--) {
        if(x[i] != y[i]) {
            return x[i] < y[i] ? -1 : 1;
        }
    }

    return 0;
}

void solve() {
    // A "corner" of size 2^j sits inside a bounding cube of side 2^{j+1} and
    // occupies 4 of the 8 unit sub-cubes meeting at one of its corners. The
    // opposite four sub-cubes form another corner, so two corners of size 2^j
    // tile a cube of side 2^{j+1}. Hence the big corner of size 2^k splits
    // into 4 cubes of side 2^k = 8 corners of size 2^{k-1}, and recursively
    // any corner of size 2^j can be replaced by 8 corners of size 2^{j-1}.
    //
    // Because of that recursion, the only constraint on which multiset of
    // small corners fills the big corner is the total volume: if x_j corners
    // of size 2^j are chosen with sum x_j * 8^j = 8^k, the tiling is always
    // realizable. The per-level cap x_j <= N_j (demand cascading from above)
    // is automatic, since sum_{i >= j} x_i * 8^i <= 8^k forces x_j to fit
    // the remaining demand. So feasibility for a chosen set S of colors is
    // just sum_{c in S, j} a[c][j] * 8^j >= 8^k.
    //
    // Minimizing the number of colors is then "smallest prefix of largest
    // volumes that reaches the target". Volumes are up to 8^1000, so each
    // color's volume is stored as a base-8 digit vector (after normalizing
    // a[c][.] by carry propagation), and colors are sorted by lexicographic
    // comparison of those digit vectors. We add colors one at a time in that
    // order and after each addition run the top-down greedy on the combined
    // per-size capacities T_j to test feasibility.
    //
    // The top-down greedy itself uses 64-bit arithmetic with a clamp on the
    // unmet demand N: as long as N stays below ~4e9 we propagate it
    // exactly; once it exceeds that bound it can no longer be absorbed by
    // any T_j (capped at c * 1000 = 1e6), so we clamp and treat the run as
    // infeasible. When feasibility holds, the loop terminates at the first
    // level where T_j >= N_j, giving the totals x_j; these are then split
    // across the chosen colors in any order, respecting a[c][j].

    vector<vector<int>> vol_digits(c);
    for(int i = 0; i < c; i++) {
        vol_digits[i] = normalize_base8(a[i]);
    }

    vector<int> order(c);
    iota(order.begin(), order.end(), 0);
    sort(order.begin(), order.end(), [&](int u, int v) {
        return cmp_digits(vol_digits[u], vol_digits[v]) > 0;
    });

    vector<int64_t> sum_caps(k, 0);
    vector<int64_t> x(k, 0);
    vector<int> chosen;
    bool feasible = false;

    auto try_greedy = [&]() -> bool {
        fill(x.begin(), x.end(), (int64_t)0);
        int64_t N = 8;
        const int64_t CLAMP = (int64_t)4e9;
        for(int j = k - 1; j >= 0; j--) {
            if(sum_caps[j] >= N) {
                x[j] = N;
                N = 0;
                break;
            }

            x[j] = sum_caps[j];
            int64_t rem = N - sum_caps[j];
            if(rem > CLAMP / 8) {
                N = CLAMP;
            } else {
                N = rem * 8;
            }
        }
        return N == 0;
    };

    for(int idx = 0; idx < c; idx++) {
        int cc = order[idx];
        chosen.push_back(cc);
        for(int j = 0; j < k; j++) {
            sum_caps[j] += a[cc][j];
        }

        if(try_greedy()) {
            feasible = true;
            break;
        }
    }

    if(!feasible) {
        cout << "NO SOLUTION\n";
        return;
    }

    vector<tuple<int, int, int>> result;
    for(int j = 0; j < k; j++) {
        int64_t need = x[j];
        for(int cc: chosen) {
            if(need == 0) {
                break;
            }

            int64_t take = min((int64_t)a[cc][j], need);
            if(take > 0) {
                result.push_back({cc + 1, j, (int)take});
                need -= take;
            }
        }
    }

    cout << result.size() << "\n";
    for(auto& [col, sz, cnt]: result) {
        cout << col << ' ' << sz << ' ' << cnt << "\n";
    }
}

int main() {
    ios_base::sync_with_stdio(false);
    cin.tie(nullptr);

    int T = 1;
    // cin >> T;
    for(int test = 1; test <= T; test++) {
        read();
        solve();
    }

    return 0;
}

=================
p480.in1
======================
1 1
8

=================
p480.ans2
======================
3
1 1 7
1 0 1
2 0 7

=================
statement.txt
======================
480. Gena's Soul Cakes
Time limit per test: 1 second(s)
Memory limit: 65536 kilobytes
input: standard
output: standard



This story is about a boy, called Gena. He is quite a usual boy who goes to school, celebrates All Hallows Eve, also known as Halloween Day, and besides likes programming contests.


One Halloween Day he went from house to house begging for "soul cakes". He happened to collect many "soul cakes" of various tastes that day, and his happiness seemed have no limits. But his elder brother Roma, who was lazy and rude boy, demanded to give him ten cakes. Gena didn't loose control over the situation and gave his brother ten cakes of the same taste in order not to show that he had many tastes, because Roma could demand more cakes. Since that time Gena was thinking about that situation. And after Gena had learnt on Geometry classes about three-dimentional space figures, he invented the following problem.


Consider non-convex figure denoted as "corner" in three-dimentional space. It consists of four equal cubes. One of the cubes has a common face with each of the others. There is the only one point belonging to all four cubes, and this point coinsides one vertex of any cubes (see picture below). Every "corner" is charaterized by two values: color and an integer length of an edge of any cubes (denoted as "size"). "Size" is always equal to some power of two, so only index is given. The problem is to construct a big "corner" comprising given figures under the restiction of using as little number of different colors as possible given "size" of this big "corner". 


Input
The first line of input contains two space separated integers: c — the number of colors, and k — the index of power of two which is equal to "size" of big "corner" (1 ≤ c, k ≤ 1000). Next c lines contain k integers aij each. aij equals the number of "corners" with color i and "size" 2j. (0 ≤ aij ≤ 1000 for 1 ≤ i ≤ c, 0 ≤ j < k).

Output
The first line of output should contain integer n denoting the number of figure types used in constructing big "corner". Next n lines should contain ai, bi, ci denoting color, "size" and number of "corner" of such type respectively. Please, separate these numbers by single spaces. If there are multiple solutions, output any of them. Every type of "corner" should appear in output no more than once. If it is impossible to construct big "corner" even using all figures, print "NO SOLUTION" to output.

Example(s)
sample input
sample output
1 1
8
1
1 0 8

sample input
sample output
2 2
1 7
7 0
3
1 1 7
1 0 1
2 0 7

=================
p480.in2
======================
2 2
1 7
7 0

=================
