////////////////////////////////////////
///  tu3 pro-con template            ///
////////////////////////////////////////
#include <cassert>
#include <cstdio>
#include <cstring>
#include <cmath>
#include <iostream>
#include <iomanip>
#include <sstream>
#include <algorithm>
#include <numeric>
#include <functional>
#include <vector>
#include <queue>
#include <string>
#include <complex>
#include <stack>
#include <set>
#include <map>
#include <list>
#include <unordered_map>
#include <unordered_set>
#include <bitset>
#include <regex>
#include <type_traits>
#include <mutex>
#include <array>
using namespace std;

//// MACRO ////
#define countof(a) (sizeof(a)/sizeof(a[0]))

#define REP(i,n) for (int i = 0; i < (n); i++)
#define RREP(i,n) for (int i = (n)-1; i >= 0; i--)
#define FOR(i,s,n) for (int i = (s); i < (n); i++)
#define RFOR(i,s,n) for (int i = (n)-1; i >= (s); i--)
#define pos(c,i) c.being() + (i)
#define allof(c) c.begin(), c.end()
#define aallof(a) a, countof(a)
#define partof(c,i,n) c.begin() + (i), c.begin() + (i) + (n)
#define apartof(a,i,n) a + (i), a + (i) + (n)
typedef unsigned int uint;
typedef long long llong;
typedef unsigned long long ullong;
#define long long long

#define EPS 1e-9
#define INF (1L << 28)
#define LINF (1LL << 60)

#define PREDICATE(t,a,exp) [&](const t & a) -> bool { return exp; }
#define COMPARISON_T(t) bool(*)(const t &, const t &)
#define COMPARISON(t,a,b,exp) [&](const t & a, const t & b) -> bool { return exp; }
#define CONVERTER(TSrc,t,TDest,exp) [&](const TSrc &t)->TDest { return exp; }

#define PRED(a,exp) [&](const auto & a) -> bool { return exp; }
#define COMP(a,b,exp) [&](const auto & a, const auto & b) -> bool { return exp; }

inline int sign_of(double x) { return abs(x) < EPS ? 0 : x > 0 ? 1 : -1; }
inline bool inRange(int val, int min, int max) { return val >= min && val < max; }
inline bool inRange(double val, double min, double max) { return val - min > -EPS && val - max < EPS; }
inline bool inRange(int x, int y, int W, int H) { return x >= 0 && x < W && y >= 0 && y < H; } // W,H含まない

template <class T> struct vevector : public vector<vector<T>> { vevector(int n = 0, int m = 0, const T &initial = T()) : vector<vector<T>>(n, vector<T>(m, initial)) { } };
template <class T> struct vevevector : public vector<vevector<T>> { vevevector(int n = 0, int m = 0, int l = 0, const T &initial = T()) : vector<vevector<T>>(n, vevector<T>(m, l, initial)) { } };
template <class T> struct vevevevector : public vector<vevevector<T>> { vevevevector(int n = 0, int m = 0, int l = 0, int k = 0, const T &initial = T()) : vector<vevevector<T>>(n, vevevector<T>(m, l, k, initial)) { } };

//// i/o helper ////

namespace std {
	template <class T1, class T2> inline istream & operator >> (istream & in, pair<T1, T2> &p) { in >> p.first >> p.second; return in; }
	template <class T1, class T2> inline ostream & operator << (ostream &out, const pair<T1, T2> &p) { out << p.first << " " << p.second; return out; }
}
template <class T> T read() { T t; cin >> t; return t; }
template <class T> vector<T> read(int n) { vector<T> v; v.reserve(n); REP(i, n) { v.push_back(read<T>()); } return v; }
template <class T> vevector<T> read(int n, int m) { vevector<T> v; REP(i, n) v.push_back(read<T>(m)); return v; }
template <class T> vector<T> readjag() { return read<T>(read<int>()); }
template <class T> vevector<T> readjag(int n) { vevector<T> v; v.reserve(n); REP(i, n) v.push_back(readjag<T>()); return v; }

template <class T> struct iter_pair_t { T beg, end; };
template <class T> iter_pair_t<T> iter_pair(T beg, T end) { return iter_pair_t<T>{beg, end}; }
template <class T> ostream & operator << (ostream &out, const iter_pair_t<T> &v)
{
	std::copy(v.beg, v.end, ostream_iterator<typename decltype(v.beg)::reference>(out, " "));
	return out;
}
template <class T1> ostream & operator << (ostream &out, const vector<T1> &v) { return out << iter_pair(begin(v), end(v)); }
template <class T1> ostream & operator << (ostream &out, const set<T1> &v) { return out << iter_pair(begin(v), end(v)); }
template <class T1, class T2> ostream & operator << (ostream &out, const map<T1, T2> &v) { return out << iter_pair(begin(v), end(v)); }

struct _Reader { istream &cin; template <class T> _Reader operator ,(T &rhs) { cin >> rhs; return *this; } };
struct _Writer { ostream &cout; bool f{ false }; template <class T> _Writer operator ,(const T &rhs) { cout << (f ? " " : "") << rhs; f = true; return *this; } };
#define READ(t,...) t __VA_ARGS__; (_Reader{cin}), __VA_ARGS__
#define WRITE(...) (_Writer{cout}), __VA_ARGS__; cout << '\n'
#define DEBUG(...) (_Writer{cerr}), __VA_ARGS__; cerr << '\n'

void solve();
int main()
{
	//freopen("input.txt", "r", stdin);
	//freopen("output.txt", "w", stdout);

	//auto temp = ctype<char>::classic_table();
	//vector<ctype<char>::mask> bar(temp, temp + ctype<char>::table_size);
	//bar[':'] |= ctype_base::space;
	//cin.imbue(locale(cin.getloc(), new ctype<char>(bar.data())));

	cin.tie(0);
	ios_base::sync_with_stdio(false);
	cout << fixed;
	cout << setprecision(std::numeric_limits<double>::max_digits10);
	solve();

	return 0;
}

////////////////////
/// template end ///
////////////////////

void solve()
{
	while (true)
	{
		READ(int, W, H, N);
		if (!N) { break; }

		{
			string dummy;
			getline(cin, dummy);
		}
		vector<string> f;
		REP(i, H)
		{
			string s;
			getline(cin, s);
			f.push_back(s);
		}

		union State
		{
			struct
			{
				uint xa : 4;
				uint ya : 4;
				uint xb : 4;
				uint yb : 4;
				uint xc : 4;
				uint yc : 4;
			};
			int pos;
		};


		vector<bool> visited(1 << 24);

		State start = {};
		State goal = {};
		{
			REP(y, H)REP(x, W)
			{
				switch(f[y][x])
				{
				case 'a': start.xa = x; start.ya = y; f[y][x] = '\0'; break;
				case 'b': start.xb = x; start.yb = y; f[y][x] = '\0'; break;
				case 'c': start.xc = x; start.yc = y; f[y][x] = '\0'; break;
				case 'A': goal.xa = x; goal.ya = y; f[y][x] = '\0'; break;
				case 'B': goal.xb = x; goal.yb = y; f[y][x] = '\0'; break;
				case 'C': goal.xc = x; goal.yc = y; f[y][x] = '\0'; break;
				case ' ': f[y][x] = '\0';  break;
				}
			}
		}
		
		queue<pair<State, int>> q;
		q.push({ start, 0 });
		visited[start.pos] = true;
		int ans = 0;
		while (!q.empty())
		{
			pair<State, int> cur = q.front(); q.pop();

			if (cur.first.pos == goal.pos)
			{
				ans = cur.second;
				break;
			}

			State p = cur.first;
			State next = {};
			int dx[] = { 0,-1, 1, 0, 0 };
			int dy[] = { 0, 0, 0,-1, 1 };
			REP(a, 5)
			{
				if (f[p.ya + dy[a]][p.xa + dx[a]]) { continue; }
				next.ya = p.ya + dy[a];
				next.xa = p.xa + dx[a];
				if (N > 1)
				{
					REP(b, 5)
					{
						if (f[p.yb + dy[b]][p.xb + dx[b]]) { continue; }
						next.yb = p.yb + dy[b];
						next.xb = p.xb + dx[b];
						if (next.xa == next.xb && next.ya == next.yb) { continue; }
						if (next.xa == p.xb && next.ya == p.yb && next.xb == p.xa && next.yb == p.ya) { continue; }
						if (N > 2)
						{
							REP(c, 5)
							{
								if (f[p.yc + dy[c]][p.xc + dx[c]]) { continue; }
								next.yc = p.yc + dy[c];
								next.xc = p.xc + dx[c];
								if (next.xa == next.xc && next.ya == next.yc) { continue; }
								if (next.xb == next.xc && next.yb == next.yc) { continue; }
								if (next.xa == p.xc && next.ya == p.yc && next.xc == p.xa && next.yc == p.ya) { continue; }
								if (next.xb == p.xc && next.yb == p.yc && next.xc == p.xb && next.yc == p.yb) { continue; }
								if (!visited[next.pos])
								{
									visited[next.pos] = true;
									q.push({ next, cur.second + 1 });
								}
							}
						}
						else
						{
							if (!visited[next.pos])
							{
								visited[next.pos] = true;
								q.push({ next, cur.second + 1 });
							}
						}
					}
				}
				else
				{
					if (!visited[next.pos])
					{
						visited[next.pos] = true;
						q.push({ next, cur.second + 1 });
					}
				}
			}
		}

		WRITE(ans);
	}
}



Measured memory_bytes: 7000