Problem: icpc2025_nac_c
Difficulty: hard
Categories: game, bit
Time limit: 5000 ms, memory limit: 2048 MB

Entrapment is an asymmetric two-player game that is played on a $3 \times 3$ square grid. The two players are called the Runner and the Trapper. The grid squares are numbered from 1 to 9 as depicted below:

<table><tr><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>23</td><td rowspan=1 colspan=1>23</td></tr><tr><td rowspan=1 colspan=1>4</td><td rowspan=1 colspan=1>5</td><td rowspan=1 colspan=1>6</td></tr><tr><td rowspan=1 colspan=1>7</td><td rowspan=1 colspan=1>89</td><td rowspan=1 colspan=1>89</td></tr></table>

Before starting the game, the players agree on the number of rounds that the game will last, and on the starting state of the game board. Up to 8 of the grid squares can be marked as unavailable. The players also choose who will be the Runner and who will be the Trapper. The Runner then secretly chooses a starting square from among those that are available (i.e., are not marked as unavailable) but does not tell the Trapper their choice.

Each of the game rounds consists of the following steps, in order:

1. The Trapper publicly chooses some subset of the available squares (the empty set is allowed) and asks the Runner, “Are you currently in any of these squares?”   
2. The Runner answers truthfully whether or not they are in any of the chosen squares.   
3. The Trapper publicly chooses exactly one available square. That square becomes unavailable for the rest of the game. (The Runner might currently reside in that square; if so, nothing special happens.)   
4. The Runner secretly moves from their current square to an available orthogonally-adjacent square. If no such square exists, the Runner announces that they are trapped and the Trapper wins the game.

If the Runner has not been trapped by the end of the last round, they prove to the Trapper that they answered all questions truthfully by revealing their choice of starting square and the move that they made during each round. The Runner then wins the game.

Because the Runner’s initial choice of square is secret, as are all of their subsequent moves, the Runner is allowed to “cheat” by not truly committing to a square. At the end of the game, if the Runner can produce a choice of starting square and subsequent moves that do not result in being trapped and are consistent with the answers to the Trapper’s questions during each round, that is enough for the Runner to win the game.

2025 ICPC North America Championship Problem C: Entrapment

# Interaction

This is an interactive problem. Given the number of game rounds and the set of squares that are initially marked unavailable, determine whether the Runner or the Trapper would win assuming optimal play, and then prove it by playing as that role against the judge. The judge will obey all game rules, but may or may not play optimally.

Interaction starts by reading a line of 2 space-separated integers $R$ and $U$ $1 \leq R \leq 9$ , $0 \leq U \leq 8$ , $R + U \le 9 )$ ): the number of rounds in the game and the number of squares that are unavailable at the start of the game.

Next, if $U > 0$ , read a line of $U$ space-separated integers $s$ ( $1 \leq s \leq 9$ ): the labels of the squares that are unavailable at the start of the game. Please refer to the diagram above for how the squares in the grid are labeled. The $U$ labels are guaranteed to be distinct.

Determine whether the Runner or Trapper would win the game with optimal play, given the starting board and number of game rounds. Print a line of output with the string Runner if the runner wins with optimal play, and the string Trapper otherwise. You will play as that role for the rest of the game; please see the appropriate section below for further instructions on how to interact with the judge in that role.

For the Runner, repeat the following steps $R$ times:

• Read a line of input with a single integer $N$ : the size of the subset of available squares that the Trapper has chosen to ask about. $N$ is guaranteed to be between 0 and the number of available squares left on the board, inclusive.   
• If $N > 0$ , read a line of $N$ space-separated integers $\ell$ $1 \leq \ell \leq 9 ,$ ) listing the labels of the squares in the Trapper’s chosen subset. The labels are guaranteed to be distinct and all of the chosen squares are guaranteed to be available.   
• Print a line of output containing either the string Yes or the string No. The former informs the trapper that you are currently in one of the chosen squares; the latter informs the trapper that you are not.   
• Read a line with a single integer $i$ $1 \leq i \leq 9$ ), the label of the square that the Trapper marks as unavailable. It is guaranteed that square $i$ is a formerly-available square.   
• Print a line with the string Free to inform the Trapper that you have secretly moved to an orthogonally-adjacent available square and are ready to proceed to the next round. If there are no orthogonally-adjacent squares available, you must print Trapped instead and exit; your submission will be judged incorrect for having failed to elude the Trapper.

After you have played $R$ rounds of the game according to the protocol above, print a line with $R + 1$ space-separated integers. The first integer is the label of your chosen starting square; each of the next $R$ integers are the labels of the squares onto which you moved at the end of each of the $R$ rounds. Your moves must be legal and must be consistent with the answers you gave to the Trapper’s queries during each round of play. After printing this line, your program must exit.

For the Trapper, repeat the following steps $R$ times:

• Print a line with a single integer $N$ : the size of the subset of available squares that you would like to ask the Runner about.   
• If $N > 0$ , print a line of $N$ space-separated integers listing the available squares to ask the Runner about. You may list the labels in any order, but the labels must be distinct and must refer to available squares.   
• Read a line of input containing a single string: Yes if the Runner is in one of your chosen squares, or $\tt N O$ otherwise.   
• Print a line with a single integer $i$ : the square that you are marking unavailable. The label $i$ must be a valid currently-available square.   
• Read a line with a single string: Free if the Runner has moved to an available square, or Trapped if they were unable to do so. After reading the word Trapped, you have won the game, and your program must exit. If you read the word Free at the end of the Rth round, your program must also exit, though your submission will be judged incorrect since you have failed to trap the Runner.

The judge is guaranteed to answer all questions truthfully.

# Notes

Do not forget to flush the output stream after each line that you print and to cleanly exit after the interaction is done. Please also make sure that you follow the above interaction protocol exactly regarding what information to print on which line of output: for example, if the protocol requires you to print a list of space-separated integers on a single line, the judge will not accept each integer on its own line.

If the judge receives invalid or unexpected input, it will print $- 1$ and then immediately exit. Your program must detect this error report and cleanly exit in order to receive a Wrong Answer verdict. If your program blocks waiting for further interaction from the judge, or tries to interpret the $- 1$ as a game move, you may receive a different rejected verdict (such as Time Limit Exceeded or Runtime Error) instead of Wrong Answer.

2025 ICPC North America Championship Problem C: Entrapment

You have been provided with a command-line tool for local testing. The tool has comments at the top to explain its use.

<table><tr><td rowspan=2 colspan=2>Read                                     Sample Interaction 1                                     Write36123789</td></tr><tr><td rowspan=1 colspan=1>36123789</td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>Trapper245</td></tr><tr><td rowspan=1 colspan=1>Yes</td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>5</td></tr><tr><td rowspan=1 colspan=1>Free</td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>0</td></tr><tr><td rowspan=1 colspan=1>No</td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>6</td></tr><tr><td rowspan=1 colspan=1>Trapped</td><td rowspan=1 colspan=1></td></tr></table>

# Sample Interaction 2

Write

<table><tr><td rowspan=2 colspan=1>20</td><td></td></tr><tr><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>Runner</td></tr><tr><td rowspan=1 colspan=1>73128945</td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>Yes</td></tr><tr><td rowspan=1 colspan=1>5</td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>Free</td></tr><tr><td rowspan=1 colspan=1>44678</td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>Yes</td></tr><tr><td rowspan=1 colspan=1>7</td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>Free541</td></tr></table>

Non-adaptive interactor (C++):
#include "testlib.h"
#include <bits/stdc++.h>

using namespace std;





static int R_game, U_game;
static vector<int> unavailable_init;
static int queries = 0;
static int query_limit = 0;


static vector<vector<int>> adj(10);







static vector<vector<vector<bool>>> dp; 


inline bool has(int mask, int i) {
    return (mask >> (i - 1)) & 1;
}


inline int with_bit(int mask, int i) {
    return mask | (1 << (i - 1));
}


inline int without_bit(int mask, int i) {
    return mask & ~(1 << (i - 1));
}



int expand_set(int p, int m) {
    int res = 0;
    for (int i = 1; i <= 9; ++i) {
        if (has(p, i)) {
            for (int nxt : adj[i]) {
                if (has(m, nxt)) {
                    res = with_bit(res, nxt);
                }
            }
        }
    }
    return res;
}





static void log_metrics() {
    try {
        tout << "queries=" << queries << endl;
        tout << "query_limit=" << query_limit << endl;
    } catch (...) {}
}

static void finish(TResult verdict, const string& msg) {
    log_metrics();
    quitf(verdict, "%s", msg.c_str());
}





void init_adj() {
    
    
    
    auto add = [&](int u, int v) {
        adj[u].push_back(v);
        adj[v].push_back(u);
    };
    
    add(1, 2); add(2, 3);
    add(4, 5); add(5, 6);
    add(7, 8); add(8, 9);
    
    add(1, 4); add(4, 7);
    add(2, 5); add(5, 8);
    add(3, 6); add(6, 9);
}

void precompute_dp() {
    dp.assign(R_game + 1, vector<vector<bool>>(512, vector<bool>(512, false)));

    
    
    
    for (int m = 0; m < 512; ++m) {
        for (int p = 0; p < 512; ++p) {
            dp[0][m][p] = (p == 0);
        }
    }

    
    for (int r = 1; r <= R_game; ++r) {
        for (int m = 0; m < 512; ++m) {
            
            
            
            
            static bool g[512];
            
            for (int s = m; ; s = (s - 1) & m) {
                bool exists_winning_block = false;
                for (int i = 1; i <= 9; ++i) {
                    if (has(m, i)) {
                        int nm = without_bit(m, i);
                        int np = expand_set(s, nm);
                        if (dp[r-1][nm][np]) {
                            exists_winning_block = true;
                            break;
                        }
                    }
                }
                
                if (s == 0) exists_winning_block = true;

                g[s] = exists_winning_block;
                if (s == 0) break;
            }

            
            
            for (int p = m; ; p = (p - 1) & m) {
                bool can_split = false;
                if (g[p]) {
                    can_split = true;
                } else {
                    for (int a = p; a > 0; a = (a - 1) & p) {
                        int b_part = p ^ a;
                        if (g[a] && g[b_part]) {
                            can_split = true;
                            break;
                        }
                    }
                }
                dp[r][m][p] = can_split;
                if (p == 0) break;
            }
        }
    }
}





void play_as_runner(int initial_mask) {
    int cur_mask = initial_mask;
    int cur_p = cur_mask;
    
    struct RoundInfo {
        int p_before;
        int q;
        string ans;
        int b;
        int p_after; 
    };
    vector<RoundInfo> history;

    for (int r = 1; r <= R_game; ++r) {
        log_metrics(); 
        
        int N;
        if (!(cin >> N)) finish(_wa, "Unexpected EOF reading query size");
        
        int Q = 0;
        if (N < 0 || N > 9) finish(_wa, "Invalid query size");
        
        for (int i=0; i<N; ++i) {
            int x;
            if (!(cin >> x)) finish(_wa, "Unexpected EOF reading query list");
            if (x < 1 || x > 9) finish(_wa, "Invalid square label");
            if (!has(cur_mask, x)) finish(_wa, "Query square not available");
            if (has(Q, x)) finish(_wa, "Duplicate square in query");
            Q = with_bit(Q, x);
        }

        
        int p_yes = cur_p & Q;
        int p_no = cur_p & (~Q); 
        int k = R_game - r; 

        
        
        
        
        
        auto is_trapper_winning = [&](int s) {
            if (s == 0) return true; 
            
            for (int i=1; i<=9; ++i) {
                if (has(cur_mask, i)) {
                    int nm = without_bit(cur_mask, i);
                    int np = expand_set(s, nm);
                    if (dp[k][nm][np]) return true;
                }
            }
            return false;
        };

        bool win_yes = is_trapper_winning(p_yes); 
        bool win_no = is_trapper_winning(p_no);   

        string my_ans;
        int next_p_initial; 

        if (!win_yes && p_yes != 0) {
            my_ans = "Yes";
            next_p_initial = p_yes;
        } else if (!win_no && p_no != 0) {
            my_ans = "No";
            next_p_initial = p_no;
        } else {
            
            if (p_yes != 0) {
                my_ans = "Yes";
                next_p_initial = p_yes;
            } else {
                my_ans = "No";
                next_p_initial = p_no;
            }
        }
        
        cout << my_ans << endl;

        int b;
        if (!(cin >> b)) finish(_wa, "Unexpected EOF reading block");
        if (b < 1 || b > 9) finish(_wa, "Invalid block label");
        if (!has(cur_mask, b)) finish(_wa, "Block square not available");
        
        int mask_after_block = without_bit(cur_mask, b);
        int p_after_move = expand_set(next_p_initial, mask_after_block);
        
        RoundInfo info;
        info.p_before = cur_p;
        info.q = Q;
        info.ans = my_ans;
        info.b = b;
        info.p_after = next_p_initial;
        history.push_back(info);

        if (p_after_move == 0) {
            cout << "Trapped" << endl;
            queries = r;
            finish(_ok, "Contestant trapped the runner");
        } else {
            cout << "Free" << endl;
            cur_p = p_after_move;
            cur_mask = mask_after_block;
            queries = r;
        }
    }

    
    vector<int> path;
    int u_curr = -1;
    for(int i=1; i<=9; ++i) if(has(cur_p, i)) { u_curr = i; break; }
    path.push_back(u_curr);

    for (int r = R_game - 1; r >= 0; --r) {
        int u_prev = -1;
        for (int cand=1; cand<=9; ++cand) {
            if (has(history[r].p_after, cand)) {
                bool adj_ok = false;
                for (int nxt : adj[cand]) {
                    if (nxt == u_curr) {
                        adj_ok = true; 
                        break;
                    }
                }
                if (adj_ok) {
                    u_prev = cand;
                    break;
                }
            }
        }
        if (u_prev == -1) finish(_fail, "Interactor internal error: path reconstruction failed");
        path.push_back(u_prev);
        u_curr = u_prev;
    }
    reverse(path.begin(), path.end());
    
    for (int i=0; i < (int)path.size(); ++i) {
        cout << path[i] << (i == (int)path.size() - 1 ? "" : " ");
    }
    cout << endl;
    
    finish(_wa, "Runner survived");
}

struct TrapHistory {
    int q_mask;
    string ans;
    int b;
};

void play_as_trapper(int initial_mask) {
    int cur_mask = initial_mask;
    int cur_p = initial_mask;
    vector<TrapHistory> hist;

    for (int r = 1; r <= R_game; ++r) {
        log_metrics();
        int k = R_game - r + 1; 
        
        int best_A = -1;
        bool can_force_win = dp[k][cur_mask][cur_p];
        
        if (can_force_win) {
            auto check_g = [&](int s) -> bool {
                if (s == 0) return true;
                for (int i=1; i<=9; ++i) {
                    if (has(cur_mask, i)) {
                        int nm = without_bit(cur_mask, i);
                        int np = expand_set(s, nm);
                        if (dp[k-1][nm][np]) return true;
                    }
                }
                return false;
            };
            for (int a = cur_p; ; a = (a-1) & cur_p) {
                if (check_g(a) && check_g(cur_p ^ a)) {
                    best_A = a;
                    break;
                }
                if (a == 0) break;
            }
        } else {
            int half_size = __builtin_popcount(cur_p) / 2;
            best_A = 0;
            for (int i=1; i<=9; ++i) {
                if (has(cur_p, i)) {
                    if (__builtin_popcount(best_A) < half_size) best_A = with_bit(best_A, i);
                }
            }
        }
        
        vector<int> q_list;
        for(int i=1; i<=9; ++i) if(has(best_A, i)) q_list.push_back(i);
        cout << q_list.size();
        for(int x : q_list) cout << " " << x;
        cout << endl;
        
        string ans;
        if (!(cin >> ans)) finish(_wa, "Unexpected EOF reading answer");
        int next_p_initial;
        if (ans == "Yes") next_p_initial = best_A;
        else if (ans == "No") next_p_initial = cur_p ^ best_A;
        else finish(_wa, "Invalid answer");
        
        int best_b = -1;
        
        for (int i=1; i<=9; ++i) {
            if (has(cur_mask, i)) {
                int nm = without_bit(cur_mask, i);
                int np = expand_set(next_p_initial, nm);
                if (dp[k-1][nm][np]) {
                    best_b = i;
                    break;
                }
            }
        }
        if (best_b == -1) {
            for(int i=1; i<=9; ++i) if(has(cur_mask, i)) { best_b = i; break; }
        }
        
        cout << best_b << endl;
        
        TrapHistory h;
        h.q_mask = best_A;
        h.ans = ans;
        h.b = best_b;
        hist.push_back(h);
        
        string res;
        if (!(cin >> res)) finish(_wa, "Unexpected EOF reading move result");
        
        if (res == "Trapped") finish(_wa, "Runner was trapped");
        else if (res == "Free") {
            cur_mask = without_bit(cur_mask, best_b);
            cur_p = expand_set(next_p_initial, cur_mask);
            queries = r;
        } else finish(_wa, "Invalid move result");
    }

    
    vector<int> path;
    for (int i=0; i <= R_game; ++i) {
        int x;
        if (!(cin >> x)) finish(_wa, "EOF reading proof");
        path.push_back(x);
    }
    
    int u = path[0];
    if (u < 1 || u > 9) finish(_wa, "Proof: Invalid start");
    for(int uv : unavailable_init) if(uv == u) finish(_wa, "Proof: Start unavailable");
    
    int check_mask = initial_mask;
    
    for (int r = 0; r < R_game; ++r) {
        bool in_q = has(hist[r].q_mask, u);
        if (hist[r].ans == "Yes" && !in_q) finish(_wa, "Proof inconsistent with answers");
        if (hist[r].ans == "No" && in_q) finish(_wa, "Proof inconsistent with answers");
        
        check_mask = without_bit(check_mask, hist[r].b);
        
        int v = path[r+1];
        if (v < 1 || v > 9) finish(_wa, "Proof: Invalid move square");
        if (!has(check_mask, v)) finish(_wa, "Proof: Move to unavailable square");
        
        bool adj_ok = false;
        for (int nxt : adj[u]) if (nxt == v) adj_ok = true;
        if (!adj_ok) finish(_wa, "Proof: Move not adjacent");
        
        u = v;
    }
    
    finish(_ok, "Correct");
}

int main(int argc, char** argv) {
    registerInteraction(argc, argv);
    atexit(log_metrics);

    
    R_game = inf.readInt();
    U_game = inf.readInt();
    query_limit = R_game;
    
    int mask = 511; 
    for (int i = 0; i < U_game; ++i) {
        int u = inf.readInt();
        unavailable_init.push_back(u);
        mask = without_bit(mask, u);
    }
    
    
    cout << R_game << " " << U_game << endl;
    if (U_game > 0) {
        for (int i = 0; i < U_game; ++i) {
            cout << unavailable_init[i] << (i == U_game - 1 ? "" : " ");
        }
        cout << endl;
    }

    init_adj();
    precompute_dp();
    
    bool trapper_wins_optimally = dp[R_game][mask][mask];
    
    string claim;
    if (!(cin >> claim)) finish(_wa, "No claim provided");
    
    bool contestant_is_runner = (claim == "Runner");
    bool contestant_is_trapper = (claim == "Trapper");
    
    if (!contestant_is_runner && !contestant_is_trapper) 
        finish(_wa, "Invalid claim (expected Runner/Trapper)");
    
    if (contestant_is_runner) {
        play_as_trapper(mask);
    } else {
        play_as_runner(mask);
    }

    return 0;
}


Adaptive interactor (C++):
#include "testlib.h"
#include <bits/stdc++.h>
using namespace std;


int R_total, U_total;
int S_initial_mask = 0;
vector<int> adj[10];




int8_t memo[10][512][512];

void init_adj() {
    
    
    
    
    auto add = [&](int u, int v) {
        adj[u].push_back(v);
        adj[v].push_back(u);
    };
    add(1, 2); add(2, 3);
    add(4, 5); add(5, 6);
    add(7, 8); add(8, 9);
    add(1, 4); add(4, 7);
    add(2, 5); add(5, 8);
    add(3, 6); add(6, 9);
}

int get_adj_mask(int u) {
    int m = 0;
    for (int v : adj[u]) m |= (1 << (v - 1));
    return m;
}


int next_p_mask(int P, int S_next) {
    int res = 0;
    for (int i = 0; i < 9; ++i) {
        if ((P >> i) & 1) { 
            
            res |= get_adj_mask(i + 1);
        }
    }
    return res & S_next;
}


bool solve(int r, int S, int P);




bool can_trapper_win_after_answer(int r, int S, int P) {
    if (P == 0) return true; 
    
    for (int i = 0; i < 9; ++i) {
        if ((S >> i) & 1) {
            int S_next = S & ~(1 << i);
            int P_next = next_p_mask(P, S_next);
            if (solve(r - 1, S_next, P_next)) return true;
        }
    }
    return false;
}


bool solve(int r, int S, int P) {
    if (P == 0) return true; 
    if (r == 0) return false; 
    
    if (memo[r][S][P] != -1) return memo[r][S][P];
    
    
    
    
    bool can_force_win = false;
    
    
    for (int A = P; ; A = (A - 1) & P) {
        if (can_trapper_win_after_answer(r, S, A)) {
            if (can_trapper_win_after_answer(r, S, P ^ A)) {
                can_force_win = true;
                break;
            }
        }
        if (A == 0) break;
    }

    return memo[r][S][P] = (can_force_win ? 1 : 0);
}


long long queries = 0;
long long query_limit = 0;

void log_metrics() {
    try {
        tout << "queries=" << queries << endl;
        tout << "query_limit=" << query_limit << endl;
    } catch (...) {}
}

void finish(TResult verdict, const string& msg) {
    log_metrics();
    quitf(verdict, "%s", msg.c_str());
}

int main(int argc, char** argv) {
    registerInteraction(argc, argv);
    atexit(log_metrics);
    
    init_adj();
    memset(memo, -1, sizeof(memo));

    
    R_total = inf.readInt();
    U_total = inf.readInt();
    S_initial_mask = (1 << 9) - 1;
    vector<int> unavailable;
    if (U_total > 0) {
        for (int i = 0; i < U_total; ++i) {
            int u = inf.readInt();
            unavailable.push_back(u);
            S_initial_mask &= ~(1 << (u - 1));
        }
    }
    
    query_limit = R_total; 
    
    
    cout << R_total << " " << U_total << endl;
    if (U_total > 0) {
        for (int i = 0; i < U_total; ++i) {
            cout << unavailable[i] << (i == U_total - 1 ? "" : " ");
        }
        cout << endl;
    }
    cout.flush();

    
    string role;
    if (!(cin >> role)) finish(_pe, "No role output");
    
    string dummy; getline(cin, dummy);

    int current_S = S_initial_mask;
    int current_P = S_initial_mask;
    
    
    struct Turn {
        int Q_mask;
        bool ans_yes;
        int u;
    };
    vector<Turn> history_runner;

    
    if (role == "Runner") {
        
        for (int r = R_total; r >= 1; --r) {
            queries++;
            
            
            int best_A = -1;
            bool winning_split_found = false;
            
            
            if (solve(r, current_S, current_P)) {
                for (int A = current_P; ; A = (A - 1) & current_P) {
                    if (can_trapper_win_after_answer(r, current_S, A) &&
                        can_trapper_win_after_answer(r, current_S, current_P ^ A)) {
                        best_A = A;
                        winning_split_found = true;
                        break;
                    }
                    if (A == 0) break;
                }
            }
            
            if (!winning_split_found) {
                
                int min_diff = 100;
                for (int A = current_P; ; A = (A - 1) & current_P) {
                    int p1 = __builtin_popcount(A);
                    int p2 = __builtin_popcount(current_P ^ A);
                    if (abs(p1 - p2) < min_diff) {
                        min_diff = abs(p1 - p2);
                        best_A = A;
                    }
                    if (A == 0) break;
                }
            }
            
            
            vector<int> Q_vec;
            for(int i=0; i<9; ++i) if((best_A >> i) & 1) Q_vec.push_back(i+1);
            cout << Q_vec.size() << endl;
            if (!Q_vec.empty()) {
                for(size_t i=0; i<Q_vec.size(); ++i) cout << Q_vec[i] << (i==Q_vec.size()-1?"":" ");
                cout << endl;
            }
            cout.flush();
            
            
            string ans;
            if (!(cin >> ans)) finish(_pe, "Expected Yes/No");
            bool is_yes = (ans == "Yes");
            if (!is_yes && ans != "No") finish(_pe, "Invalid answer string");
            
            if (is_yes) current_P &= best_A;
            else current_P &= ~best_A;
            
            
            int best_u = -1;
            bool winning_u_found = false;
            
            
            for (int i=0; i<9; ++i) {
                if ((current_S >> i) & 1) {
                    int S_next = current_S & ~(1 << i);
                    int P_next = next_p_mask(current_P, S_next);
                    if (solve(r-1, S_next, P_next)) {
                        best_u = i+1;
                        winning_u_found = true;
                        break;
                    }
                }
            }
            
            if (!winning_u_found) {
                
                int min_sz = 100;
                for (int i=0; i<9; ++i) {
                    if ((current_S >> i) & 1) {
                        int S_next = current_S & ~(1 << i);
                        int P_next = next_p_mask(current_P, S_next);
                        int sz = __builtin_popcount(P_next);
                        if (sz < min_sz) {
                            min_sz = sz;
                            best_u = i+1;
                        }
                    }
                }
            }
            
            cout << best_u << endl;
            cout.flush();

            
            history_runner.push_back({best_A, is_yes, best_u});
            
            
            current_S &= ~(1 << (best_u - 1));
            current_P = next_p_mask(current_P, current_S);
            
            
            string status;
            if (!(cin >> status)) finish(_pe, "Expected Free/Trapped");
            
            if (status == "Trapped") {
                if (current_P == 0) finish(_wa, "Contestant claimed Runner but was trapped (correctly)");
                else finish(_wa, "Contestant claimed Runner but gave up (Trapped)");
            } else if (status != "Free") {
                finish(_pe, "Invalid status string");
            }
        }
        
        
        vector<int> proof(R_total + 1);
        for(int i=0; i<=R_total; ++i) {
            if (!(cin >> proof[i])) finish(_pe, "Expected proof integer");
        }
        
        int current_pos = proof[0];
        if (current_pos < 1 || current_pos > 9 || !((S_initial_mask >> (current_pos-1)) & 1))
            finish(_wa, "Proof: Invalid starting square");

        int avail_mask = S_initial_mask;
        for(int k=0; k<R_total; ++k) {
            
            bool in_Q = (history_runner[k].Q_mask >> (current_pos-1)) & 1;
            if (history_runner[k].ans_yes && !in_Q) finish(_wa, "Proof: Pos not in Query but answered Yes");
            if (!history_runner[k].ans_yes && in_Q) finish(_wa, "Proof: Pos in Query but answered No");
            
            
            int u = history_runner[k].u;
            avail_mask &= ~(1 << (u-1));
            
            int next_pos = proof[k+1];
            if (next_pos < 1 || next_pos > 9) finish(_wa, "Proof: Invalid square");
            if (!((avail_mask >> (next_pos-1)) & 1)) finish(_wa, "Proof: Move to unavailable square");
            
            
            bool adj_ok = false;
            for(int nb : adj[current_pos]) if(nb == next_pos) adj_ok = true;
            if (!adj_ok) finish(_wa, "Proof: Move not adjacent");
            
            current_pos = next_pos;
        }
        finish(_ok, "Survived and proved");
        
    } else if (role == "Trapper") {
        
        struct Step {
            int P_after_ans; 
            int removed;     
        };
        vector<Step> history;
        
        for (int r = R_total; r >= 1; --r) {
            queries++;
            
            
            int N; 
            if (!(cin >> N)) finish(_pe, "Expected N");
            int Q_mask = 0;
            for(int i=0; i<N; ++i) {
                int x; 
                if (!(cin >> x)) finish(_pe, "Expected x");
                Q_mask |= (1 << (x-1));
            }
            
            
            int P_yes = current_P & Q_mask;
            int P_no = current_P & ~Q_mask;
            
            string my_ans;
            
            bool win_yes = solve(r, current_S, P_yes);
            bool win_no = solve(r, current_S, P_no);
            
            if (!win_yes && !win_no) {
                if (__builtin_popcount(P_yes) >= __builtin_popcount(P_no)) my_ans = "Yes";
                else my_ans = "No";
            } else if (!win_yes) {
                my_ans = "Yes";
            } else if (!win_no) {
                my_ans = "No";
            } else {
                if (__builtin_popcount(P_yes) >= __builtin_popcount(P_no)) my_ans = "Yes";
                else my_ans = "No";
            }
            
            if (my_ans == "Yes") current_P = P_yes;
            else current_P = P_no;
            
            cout << my_ans << endl;
            cout.flush();
            
            int P_after_answer = current_P;
            
            
            int u; 
            if (!(cin >> u)) finish(_pe, "Expected removal");
            current_S &= ~(1 << (u-1));
            
            
            current_P = next_p_mask(current_P, current_S);
            
            history.push_back({P_after_answer, u});
            
            if (current_P == 0) {
                cout << "Trapped" << endl;
                cout.flush();
                finish(_ok, "Interactor trapped (Contestant won)");
            } else {
                cout << "Free" << endl;
                cout.flush();
            }
        }
        
        
        vector<int> path(R_total + 1);
        int curr = -1;
        for(int i=0; i<9; ++i) if((current_P >> i) & 1) { curr = i+1; break; }
        path[R_total] = curr;
        
        for (int k = R_total - 1; k >= 0; --k) {
            int prev_P = history[k].P_after_ans;
            
            int found = -1;
            for(int i=0; i<9; ++i) {
                if ((prev_P >> i) & 1) {
                    int p = i+1;
                    for (int nb : adj[p]) if (nb == path[k+1]) found = p;
                }
            }
            path[k] = found;
        }
        
        for (int i=0; i<=R_total; ++i) cout << path[i] << (i==R_total?"":" ");
        cout << endl;
        finish(_wa, "Interactor survived (Contestant lost)");
        
    } else {
        finish(_pe, "Invalid role string");
    }

    return 0;
}


Test generator (C++):
#include "testlib.h"
#include <bits/stdc++.h>

using namespace std;

int main(int argc, char* argv[]) {
    registerGen(argc, argv, 1);

    
    
    
    string mode = opt<string>("mode", "non");

    
    int seed_idx = -1;
    if (argc > 1) {
        try {
            seed_idx = stoi(argv[1]);
        } catch (...) {
            seed_idx = -1;
        }
    }

    int R, U;
    vector<int> unavailable;
    vector<int> grid(9);
    iota(grid.begin(), grid.end(), 1);

    
    if (seed_idx == 1) {
        
        
        R = 9;
        U = 0;
    } else if (seed_idx == 2) {
        
        
        R = 1;
        U = 8;
        shuffle(grid.begin(), grid.end());
        for (int i = 0; i < U; ++i) unavailable.push_back(grid[i]);
    } else if (seed_idx == 3) {
        
        
        
        U = 4;
        R = 5; 
        unavailable = {2, 4, 6, 8};
    } else {
        
        
        
        
        U = rnd.next(0, 8);
        
        
        
        int max_R = 9 - U;
        if (rnd.next(3) == 0) {
            R = max_R;
        } else {
            R = rnd.next(1, max_R);
        }

        
        shuffle(grid.begin(), grid.end());
        for (int i = 0; i < U; ++i) {
            unavailable.push_back(grid[i]);
        }
    }

    sort(unavailable.begin(), unavailable.end());

    
    
    
    if (mode == "adp") {
        cout << R << " " << U << endl;
        if (U > 0) {
            for (int i = 0; i < U; ++i) {
                cout << unavailable[i] << (i == U - 1 ? "" : " ");
            }
            cout << endl;
        }
    } else {
        
        cout << R << " " << U << endl;
        if (U > 0) {
            for (int i = 0; i < U; ++i) {
                cout << unavailable[i] << (i == U - 1 ? "" : " ");
            }
            cout << endl;
        }
    }

    return 0;
}