#include <iostream>
#include <vector>
#include <string>
#include <queue>
// #include <set> // No longer needed
#include <unordered_set> // Use unordered_set for visited states
#include <map>
#include <tuple> // For std::tie used in comparison
#include <vector>
#include <algorithm> // For std::sort
#include <cctype> // For std::isdigit
#include <functional> // For std::hash
using namespace std;
// Struct to represent a robot (original or composite)
struct Robot {
int min_l, max_l; // Min and max labels of merged robots
int r, c; // Row and column position
// Comparison operator for sorting (needed for canonical state representation)
bool operator<(const Robot& other) const {
return tie(min_l, max_l, r, c) < tie(other.min_l, other.max_l, other.r, other.c);
}
// Equality operator (needed for unordered_set)
bool operator==(const Robot& other) const {
return tie(min_l, max_l, r, c) == tie(other.min_l, other.max_l, other.r, other.c);
}
};
// --- Hashing Implementation ---
// Helper function to combine hashes (based on boost::hash_combine)
template <class T>
inline void hash_combine(size_t& seed, const T& v) {
hash<T> hasher;
seed ^= hasher(v) + 0x9e3779b9 + (seed << 6) + (seed >> 2);
}
// Hash function for Robot struct
struct RobotHash {
size_t operator()(const Robot& r) const {
size_t seed = 0;
hash_combine(seed, r.min_l);
hash_combine(seed, r.max_l);
hash_combine(seed, r.r);
hash_combine(seed, r.c);
return seed;
}
};
// Hash function for vector<Robot>
// IMPORTANT: Assumes the vector<Robot> is already sorted into canonical form!
struct VectorRobotHash {
size_t operator()(const vector<Robot>& robots) const {
size_t seed = robots.size(); // Start seed with size
RobotHash robot_hasher;
for(const auto& r : robots) {
// Combine hash of each robot into the seed
seed ^= robot_hasher(r) + 0x9e3779b9 + (seed << 6) + (seed >> 2);
}
return seed;
}
};
// --- Constants and Global Variables ---
const int MAX_H = 501;
const int MAX_W = 501;
int N;
int W;
int H;
vector<string> grid;
vector<pair<int, int>> initial_pos;
int dr[] = {-1, 0, 1, 0};
int dc[] = {0, 1, 0, -1};
pair<int, int> precomputed_moves[MAX_H][MAX_W][4];
// Function to check if a cell is valid
bool isValid(int r, int c) {
return r >= 0 && r < H && c >= 0 && c < W && grid[r][c] != 'x';
}
// Precomputation Function (No changes needed here)
void precompute_all_moves() {
for (int r = 0; r < H; ++r) {
for (int c = 0; c < W; ++c) {
if (grid[r][c] == 'x') {
for (int start_dir = 0; start_dir < 4; ++start_dir) precomputed_moves[r][c][start_dir] = {-1, -1};
continue;
}
for (int start_dir = 0; start_dir < 4; ++start_dir) {
int nr = r, nc = c;
int current_dir = start_dir;
bool starting_on_rotator = (grid[r][c] == 'A' || grid[r][c] == 'C');
int first_step_dir = start_dir;
if (starting_on_rotator) {
int rotated_dir = (grid[r][c] == 'A') ? (start_dir + 3) % 4 : (start_dir + 1) % 4;
int rotated_next_r = r + dr[rotated_dir];
int rotated_next_c = c + dc[rotated_dir];
if (isValid(rotated_next_r, rotated_next_c)) {
first_step_dir = rotated_dir;
}
}
int first_step_r = r + dr[first_step_dir];
int first_step_c = c + dc[first_step_dir];
if (!isValid(first_step_r, first_step_c)) {
precomputed_moves[r][c][start_dir] = {r, c};
continue;
}
nr = first_step_r; nc = first_step_c;
current_dir = first_step_dir;
while (true) {
if (grid[nr][nc] == 'A') current_dir = (current_dir + 3) % 4;
else if (grid[nr][nc] == 'C') current_dir = (current_dir + 1) % 4;
int next_r = nr + dr[current_dir];
int next_c = nc + dc[current_dir];
if (!isValid(next_r, next_c)) break;
nr = next_r; nc = next_c;
}
precomputed_moves[r][c][start_dir] = {nr, nc};
}
}
}
}
// Optimized simulate_move using precomputation
pair<int, int> simulate_move(int r, int c, int dir) {
return precomputed_moves[r][c][dir];
}
// Function to perform merges (No changes needed here)
// IMPORTANT: This function sorts the vector at the end, which is required for hashing.
void perform_merges(vector<Robot>& current_robots) {
if (current_robots.size() < 2) return;
bool merged_in_pass = true;
while (merged_in_pass) {
merged_in_pass = false;
vector<Robot> next_round_robots;
vector<bool> processed(current_robots.size(), false);
// Sort by position first to efficiently find robots at the same location
sort(current_robots.begin(), current_robots.end(), [](const Robot& a, const Robot& b){
return tie(a.r, a.c, a.min_l, a.max_l) < tie(b.r, b.c, b.min_l, b.max_l);
});
for (int i = 0; i < current_robots.size(); ++i) {
if (processed[i]) continue;
bool merged_robot_i = false;
// Check subsequent robots ONLY if they are at the same location
for (int j = i + 1; j < current_robots.size(); ++j) {
if (processed[j]) continue;
// Optimization: If position differs, no more robots at location i can be found
if (current_robots[i].r != current_robots[j].r || current_robots[i].c != current_robots[j].c) break;
// Check compatibility
if (current_robots[i].max_l + 1 == current_robots[j].min_l || current_robots[j].max_l + 1 == current_robots[i].min_l) {
Robot merged_robot;
merged_robot.min_l = min(current_robots[i].min_l, current_robots[j].min_l);
merged_robot.max_l = max(current_robots[i].max_l, current_robots[j].max_l);
merged_robot.r = current_robots[i].r;
merged_robot.c = current_robots[i].c;
next_round_robots.push_back(merged_robot);
processed[i] = true; processed[j] = true;
merged_in_pass = true; merged_robot_i = true;
break; // Robot i merged, move to next i
}
}
// If robot i wasn't processed (didn't merge), add it back
if (!processed[i]) {
next_round_robots.push_back(current_robots[i]);
processed[i] = true; // Mark as processed (kept)
}
}
current_robots = next_round_robots; // Update for next pass
if (current_robots.size() < 2) break; // Stop if few robots left
}
// Final sort by labels/pos for canonical representation before hashing/storing
sort(current_robots.begin(), current_robots.end());
}
// Solves the problem using BFS with precomputed moves and hashing
int solve() {
vector<Robot> initial_state;
for (int i = 0; i < N; ++i) {
initial_state.push_back({i + 1, i + 1, initial_pos[i].first, initial_pos[i].second});
}
perform_merges(initial_state); // Sorts the state canonically
queue<pair<vector<Robot>, int>> q;
// *** Use unordered_set with custom hash function ***
unordered_set<vector<Robot>, VectorRobotHash> visited;
q.push({initial_state, 0});
visited.insert(initial_state); // Insert requires state to be sorted
while (!q.empty()) {
vector<Robot> current_robots = q.front().first; // Copy state
int current_pushes = q.front().second;
q.pop();
// Goal check
if (current_robots.size() == 1 && current_robots[0].min_l == 1 && current_robots[0].max_l == N) {
return current_pushes;
}
// Generate next states
for (int i = 0; i < current_robots.size(); ++i) {
for (int dir_idx = 0; dir_idx < 4; ++dir_idx) {
// It's often faster to modify a copy than to create a new vector each time
vector<Robot> next_robots = current_robots;
Robot moving_robot = next_robots[i]; // Get robot to move (before potential modification)
pair<int, int> final_pos = simulate_move(moving_robot.r, moving_robot.c, dir_idx);
if (final_pos.first == -1) continue; // Invalid start pos for move
// Update the specific robot that moved
next_robots[i].r = final_pos.first;
next_robots[i].c = final_pos.second;
// Perform merges. This sorts the vector `next_robots` canonically.
perform_merges(next_robots);
// Check visited using unordered_set's find (average O(1) + hashing cost)
if (visited.find(next_robots) == visited.end()) {
visited.insert(next_robots); // Insert requires state to be sorted
q.push({next_robots, current_pushes + 1});
}
}
}
}
return -1; // Goal not reachable
}
int main() {
ios_base::sync_with_stdio(false);
cin.tie(NULL);
cin >> N >> W >> H;
grid.resize(H);
initial_pos.resize(N);
vector<bool> pos_found(N, false);
for (int i = 0; i < H; ++i) {
cin >> grid[i];
for (int j = 0; j < W; ++j) {
if (isdigit(grid[i][j])) {
int robot_label = grid[i][j] - '0';
int robot_id = robot_label - 1;
if (robot_id >= 0 && robot_id < N) {
initial_pos[robot_id] = {i, j};
pos_found[robot_id] = true;
grid[i][j] = '.';
} else {
grid[i][j] = '.';
}
}
}
}
bool all_found = true;
for(int i=0; i<N; ++i) {
if (!pos_found[i]) {
all_found = false;
cerr << "Error: Robot " << (i+1) << " not found on the initial grid." << endl;
return 1;
}
}
precompute_all_moves();
cout << solve() << endl;
return 0;
}
