#include using namespace std; int n; long long k; vector memo; int query_count = 0; long long do_query(int r, int c) { int key = r * 2001 + c; if (memo[key] != -1) return memo[key]; cout << "QUERY " << r << " " << c << "\n"; cout.flush(); long long v; cin >> v; memo[key] = v; query_count++; return v; } void done(long long ans) { cout << "DONE " << ans << "\n"; cout.flush(); exit(0); } // Staircase walk: count elements <= x, track boundary per row // posOut[i] = max column j such that a[i][j] <= x (0 if none) // Walk top-to-bottom, right-to-left: O(n) new queries (with caching, even less) long long count_leq(long long x, vector& pos) { pos.assign(n + 1, 0); long long cnt = 0; int j = n; for (int i = 1; i <= n; i++) { while (j >= 1 && do_query(i, j) > x) j--; pos[i] = j; cnt += j; if (j == 0) { for (int ii = i + 1; ii <= n; ii++) pos[ii] = 0; break; } } return cnt; } long long count_lt(long long x, vector& pos) { pos.assign(n + 1, 0); long long cnt = 0; int j = n; for (int i = 1; i <= n; i++) { while (j >= 1 && do_query(i, j) >= x) j--; pos[i] = j; cnt += j; if (j == 0) { for (int ii = i + 1; ii <= n; ii++) pos[ii] = 0; break; } } return cnt; } // Bounded staircase walk: only check within posLow[i]+1..posHigh[i] // More query-efficient when bounds are tight long long count_leq_bounded(long long x, vector& pos, const vector& lo, const vector& hi) { pos.assign(n + 1, 0); long long cnt = 0; // For row i, we know answer is in [lo[i]+1, hi[i]] // Walk top to bottom. j starts from hi[1] and decreases. int j = n; // will be clamped for (int i = 1; i <= n; i++) { j = min(j, hi[i]); if (j < 1) { pos[i] = 0; for (int ii = i + 1; ii <= n; ii++) pos[ii] = 0; break; } while (j >= 1 && do_query(i, j) > x) j--; pos[i] = j; cnt += max(0, j); if (j == 0) { for (int ii = i + 1; ii <= n; ii++) pos[ii] = 0; break; } } return cnt; } void solve() { long long N2 = (long long)n * n; if (n == 1) { done(do_query(1, 1)); return; } if (k == 1) { done(do_query(1, 1)); return; } if (k == N2) { done(do_query(n, n)); return; } // For small k or k close to N2, use heap directly long long heap_k = min(k, N2 - k + 1); if (heap_k + n <= 24000) { if (k <= N2 - k + 1) { // k-th smallest via min-heap on rows priority_queue, vector>, greater<>> pq; for (int i = 1; i <= n; i++) pq.emplace(do_query(i, 1), i, 1); long long result = -1; for (long long t = 0; t < k; t++) { auto [v, r, c] = pq.top(); pq.pop(); result = v; if (c + 1 <= n) pq.emplace(do_query(r, c + 1), r, c + 1); } done(result); } else { long long kk = N2 - k + 1; priority_queue> pq; for (int i = 1; i <= n; i++) pq.emplace(do_query(i, n), i, n); long long result = -1; for (long long t = 0; t < kk; t++) { auto [v, r, c] = pq.top(); pq.pop(); result = v; if (c - 1 >= 1) pq.emplace(do_query(r, c - 1), r, c - 1); } done(result); } } // Value-based narrowing approach mt19937_64 rng(12345); // Initialize: posHigh from count_leq(a[n][n]), posLow all 0 vector posHigh(n + 1, n), posLow(n + 1, 0); long long cntHigh = N2, cntLow = 0; long long highTh, lowTh; // Get initial bounds - use anti-diagonal estimate // Row ceil(k/n): a[ceil(k/n)][n] is an upper bound int rBound = (int)min((long long)n, (k + n - 1) / n); highTh = do_query(rBound, n); cntHigh = count_leq(highTh, posHigh); if (cntHigh < k) { highTh = do_query(n, n); cntHigh = count_leq(highTh, posHigh); } // Lower bound: a[floor(k/n)+1][1] or just 0 // Actually, row floor(k/n): a[floor(k/n)][1] has at most floor(k/n)*n elements <= it from first floor(k/n) rows // But it's more complex. Just use cntLow = 0 initially. lowTh = do_query(1, 1) - 1; // everything is >= a[1][1] // posLow stays all 0 const int SAMPLES = 15; for (int iter = 0; iter < 200; iter++) { if (cntHigh < k) { highTh = do_query(n, n); cntHigh = count_leq(highTh, posHigh); if (cntHigh < k) break; // shouldn't happen } long long candTotal = cntHigh - cntLow; if (candTotal <= 0) { done(highTh); } long long needSmall = k - cntLow; long long needLarge = cntHigh - k + 1; // Count non-empty rows and build prefix sums for sampling vector pref(n + 1, 0); int nonempty = 0; for (int i = 1; i <= n; i++) { int len = posHigh[i] - posLow[i]; if (len > 0) nonempty++; pref[i] = pref[i - 1] + max(0, len); } candTotal = pref[n]; if (candTotal <= 0) { done(highTh); } // Check if we can enumerate directly long long budget = 49500 - query_count; long long enumCost = min(needSmall, needLarge) + nonempty + 10; if (enumCost <= budget) { // Enumerate via heap if (needSmall <= needLarge) { priority_queue, vector>, greater<>> pq; for (int i = 1; i <= n; i++) { int L = posLow[i] + 1; int R = posHigh[i]; if (L >= 1 && L <= n && L <= R) { pq.emplace(do_query(i, L), i, L); } } long long result = 0; for (long long t = 0; t < needSmall; t++) { auto [v, r, c] = pq.top(); pq.pop(); result = v; if (c + 1 <= posHigh[r]) pq.emplace(do_query(r, c + 1), r, c + 1); } done(result); } else { priority_queue> pq; for (int i = 1; i <= n; i++) { int L = posLow[i] + 1; int R = posHigh[i]; if (R >= 1 && R <= n && L <= R) { pq.emplace(do_query(i, R), i, R); } } long long result = 0; for (long long t = 0; t < needLarge; t++) { auto [v, r, c] = pq.top(); pq.pop(); result = v; int L = posLow[r] + 1; if (c - 1 >= L) pq.emplace(do_query(r, c - 1), r, c - 1); } done(result); } } // Budget check for another iteration if (budget < 2 * n + SAMPLES + 200) { // Try to enumerate anyway if (nonempty + min(needSmall, needLarge) + 10 <= budget) { // same enumeration as above if (needSmall <= needLarge) { priority_queue, vector>, greater<>> pq; for (int i = 1; i <= n; i++) { int L = posLow[i] + 1, R = posHigh[i]; if (L >= 1 && L <= n && L <= R) pq.emplace(do_query(i, L), i, L); } long long result = 0; for (long long t = 0; t < needSmall; t++) { auto [v, r, c] = pq.top(); pq.pop(); result = v; if (c + 1 <= posHigh[r]) pq.emplace(do_query(r, c + 1), r, c + 1); } done(result); } else { priority_queue> pq; for (int i = 1; i <= n; i++) { int L = posLow[i] + 1, R = posHigh[i]; if (R >= 1 && R <= n && L <= R) pq.emplace(do_query(i, R), i, R); } long long result = 0; for (long long t = 0; t < needLarge; t++) { auto [v, r, c] = pq.top(); pq.pop(); result = v; if (c - 1 >= posLow[r] + 1) pq.emplace(do_query(r, c - 1), r, c - 1); } done(result); } } done(highTh); // fallback } // Sample pivot values from candidate segments vector samp; samp.reserve(SAMPLES); for (int s = 0; s < SAMPLES; s++) { long long pick = 1 + rng() % candTotal; int row = (int)(lower_bound(pref.begin() + 1, pref.end(), pick) - pref.begin()); if (row < 1 || row > n) continue; int len = posHigh[row] - posLow[row]; if (len <= 0) { continue; } int col = posLow[row] + 1 + rng() % len; col = max(1, min(n, col)); samp.push_back(do_query(row, col)); } if (samp.empty()) { done(highTh); } sort(samp.begin(), samp.end()); // Pick pivot at the quantile matching needSmall/candTotal double q = (double)needSmall / (double)candTotal; int idx = (int)(q * (double)(samp.size() - 1)); idx = max(0, min((int)samp.size() - 1, idx)); long long pivot = samp[idx]; // Count elements <= pivot vector posPivot; long long cntPivot = count_leq(pivot, posPivot); if (cntPivot >= k) { // Tighten high if (pivot == highTh && cntPivot == cntHigh) { // Pivot equals current high and same count - duplicates vector posLess; long long cntLess = count_lt(pivot, posLess); if (cntLess < k) { done(pivot); // k-th element equals pivot } // Narrow to strictly less than pivot highTh = pivot - 1; posHigh = posLess; cntHigh = cntLess; } else { highTh = pivot; posHigh = posPivot; cntHigh = cntPivot; } } else { // Tighten low lowTh = pivot; posLow = posPivot; cntLow = cntPivot; } } } int main() { ios_base::sync_with_stdio(false); cin.tie(nullptr); cin >> n >> k; memo.assign(2002 * 2002, -1); solve(); return 0; }