#include <cstdio>
#include <iostream>
#include <queue>
#include <set>
#include <utility>

using namespace std;

int w, h, n;
char field[17][17];

static int dx[5] = {0, -1, 0, 1, 0};
static int dy[5] = {0, 0, 1, 0, -1};

#define ENCODE(ax,ay,bx,by,cx,cy) (((ax)<<0)|((ay)<<4)|((bx)<<8)|((by)<<12)|((cx)<<16)|((cy)<<20))

int main()
{
  while (cin >> w >> h >> n) {
    if (w == 0 && h == 0 && n == 0) { break; }
    cin.ignore(w+1, '\n');

    int start_state = 0x000000, goal_state = 0x000000;
    for (int i = 0; i < h; ++i) {
      cin.get(field[i], w+1, '\n'); cin.ignore(w+1, '\n');
      for (int j = 0; j < w; ++j) {
        if (field[i][j] == 'a') { start_state |= (j <<  0) | (i <<  4); }
        if (field[i][j] == 'b') { start_state |= (j <<  8) | (i << 12); }
        if (field[i][j] == 'c') { start_state |= (j << 16) | (i << 20); }
        if (field[i][j] == 'A') { goal_state  |= (j <<  0) | (i <<  4); }
        if (field[i][j] == 'B') { goal_state  |= (j <<  8) | (i << 12); }
        if (field[i][j] == 'C') { goal_state  |= (j << 16) | (i << 20); }
      }
    }

    queue< pair<int, int> > current_states;
    set<int> visited_states;
    current_states.push(make_pair(start_state, 1));
    visited_states.insert(start_state);
    while (!current_states.empty()) {
      pair<int, int> p = current_states.front();
      current_states.pop();
      int steps = p.second;
      int ax = (p.first & 0x00000F) >>  0;
      int ay = (p.first & 0x0000F0) >>  4;
      int bx = (p.first & 0x000F00) >>  8;
      int by = (p.first & 0x00F000) >> 12;
      int cx = (p.first & 0x0F0000) >> 16;
      int cy = (p.first & 0xF00000) >> 20;

      for (int ai = 0; ai < 5; ++ai) {
	int nax = ax + dx[ai], nay = ay + dy[ai];
	if (nax < 0 || nax >= w || nay < 0 || nay >= h || field[nay][nax] == '#') { continue; }
	for (int bi = 0; bi < (n >= 2 ? 5 : 1); ++bi) {
	  int nbx = bx + dx[bi], nby = by + dy[bi];
	  if (nbx < 0 || nbx >= w || nby < 0 || nby >= h ||
	      (n >= 2 && field[nby][nbx] == '#')) { continue; }
	  if (n >= 2 && nax == nbx && nay == nby) { continue; }
	  if (n >= 2 && nax == bx && nay == by && nbx == ax && nby == ay) { continue; }
	  for (int ci = 0; ci < (n >= 3 ? 5 : 1); ++ci) {
	    int ncx = cx + dx[ci], ncy = cy + dy[ci];
	    if (ncx < 0 || ncx >= w || ncy < 0 || ncy >= h ||
		(n >= 3 && field[ncy][ncx] == '#')) { continue; }
	    if (n >= 3 && nax == ncx && nay == ncy) { continue; }
	    if (n >= 3 && nbx == ncx && nby == ncy) { continue; }
	    if (n >= 3 && nax == cx && nay == cy && ncx == ax && ncy == ay) { continue; }
	    if (n >= 3 && ncx == bx && ncy == by && nbx == cx && nby == cy) { continue; }
	    int new_state = ENCODE(nax, nay, nbx, nby, ncx, ncy);
	    if (new_state == goal_state) {
	      cout << steps << endl;
	      goto found;
	    }
	    if (visited_states.find(new_state) == visited_states.end()) {
	      current_states.push(make_pair(new_state, steps+1));
	      visited_states.insert(new_state);
	    }
	  }
	}
      }
    }
  found:
    ;
  }
  return 0;
}