#include <iostream>
#include <sstream>
#include <string>
#include <algorithm>
#include <vector>
#include <stack>
#include <queue>
#include <set>
#include <map>
#include <cstdio>
#include <cstdlib>
#include <cstring>
#include <cmath>
#include <cassert>
 
using namespace std;
 
#define FOR(i,k,n) for(int i=(k); i<(int)(n); ++i)
#define REP(i,n) FOR(i,0,n)
#define FORIT(i,c) for(__typeof((c).begin())i=(c).begin();i!=(c).end();++i)
 
template<class T> void debug(T begin, T end){ for(T i = begin; i != end; ++i) cerr<<*i<<" "; cerr<<endl; }
 
typedef long long ll;
const int INF = 100000000;
const double EPS = 1e-8;
const int MOD = 1000000007;
int dx[5] = {1, 0, -1, 0, 0};
int dy[5] = {0, 1, 0, -1, 0};
typedef pair<int, int> P;
const int MAX_L = 16;
const int MAX_N = 3;
struct State{
  P p[MAX_N];
  int d, f;
  State(P pp[MAX_N], int dd, int hh){
    REP(i, MAX_N) p[i] = pp[i];
    d = dd;
    f = dd + hh;
  }
  bool operator < (const State& s) const {
    if(f != s.f) return f > s.f;
    return d < s.d;
  }
};
int W, H, N;
string grid[MAX_L];
inline int make(const P p[MAX_N]){
  int res = 0;
  REP(i, N) {
    res = res << 8 | (p[i].first << 4) | p[i].second;
  }
  return res;
}
unsigned short dist[1 << 24];
inline void add(const State& s, priority_queue<State>& que){
  int k = make(s.p);
  if(dist[k] > s.d){
    dist[k] = s.d;
    que.push(s);
  }
}
inline bool equal(P p[MAX_N], P q[MAX_N]){
  REP(i, N) if(p[i] != q[i]) return false;
  return true;
}
inline bool valid(int x, int y, int W, int H){
  return (x >= 0 && y >= 0 && x < W && y < H) && grid[y][x] != '#';
}
inline bool valid(const State& s, const State& prev){
  REP(i, N) if(!valid(s.p[i].first, s.p[i].second, W, H)) return false;
  FOR(i, 0, N) FOR(j, i + 1, N) if(s.p[i] == s.p[j]) return false;
  FOR(i, 0, N) FOR(j, i + 1, N) if(s.p[i] == prev.p[j] && s.p[j] == prev.p[i]) return false;
  return true;
}
int memo[16][16][16][16];
int calc_distance(P start, P goal){
  int& dist = memo[start.first][start.second][goal.first][goal.second];
  if(dist != -1) return dist;
  queue<P> que;
  que.push(goal);
  memo[goal.first][goal.second][goal.first][goal.second] = 0;
  while(!que.empty()){
    P p = que.front(); que.pop();
    REP(r, 4){
      P q = p;
      q.first += dx[r];
      q.second += dy[r];
      if(!valid(q.first, q.second, W, H)) continue;
      if(memo[q.first][q.second][goal.first][goal.second] != -1) continue;
      memo[q.first][q.second][goal.first][goal.second] = memo[p.first][p.second][goal.first][goal.second] + 1;
      que.push(q);
    }
  }
  return dist;
}
void print(State s){
  printf("d = %d h = %d f = %d ", s.d, s.f - s.d, s.f);
  REP(i, N) printf("(%d, %d) ", s.p[i].first, s.p[i].second);
  cout << endl;
}
int main(){
  while(cin >> W >> H >> N && N){
    memset(memo, -1, sizeof memo);
    REP(i, 1 << 24) dist[i] = 16 * 16 * 3;
    cin.ignore();
    REP(y, H) getline(cin, grid[y]);
    P goal[MAX_N];
    P start[MAX_N];
    REP(y, H)REP(x, W){
      if(islower(grid[y][x])){
        int k = grid[y][x] - 'a';
        grid[y][x] = ' ';
        start[k] = P(x, y);
      }
      if(isupper(grid[y][x])){
        int k = grid[y][x] - 'A';
        grid[y][x] = ' ';
        goal[k] = P(x, y);
      }
    }
    priority_queue<State> que;
    State init(start, 0, 1);
    add(init, que);
    while(!que.empty()){
      State s = que.top(); que.pop();
      //print(s);
      if(s.d == s.f){
        cout << s.d << endl;
        break;
      }
      REP(SR, 5 * 5 * 5){
        int R = SR;
        State next = s;
        REP(i, N){
          int r = R % 5;
          R /= 5;
          next.p[i].first += dx[r];
          next.p[i].second += dy[r];
        }
        if(!valid(next, s)) continue;
        next.d ++;
        int h = 0;
        REP(i, N){
          h = max(h, calc_distance(next.p[i], goal[i]));
        }
        next.f = next.d + h;
        assert(h < INF);
        add(next, que);
      }
    }
  }
  return 0;
}

Measured memory_bytes: 35252