Problem:
Score : 700 points

Problem Statement

Takahashi is locked within a building.

This building consists of H×W rooms, arranged in H rows and W columns.
We will denote the room at the i-th row and j-th column as (i,j). The state of this room is represented by a character A_{i,j}. If A_{i,j}= #, the room is locked and cannot be entered; if A_{i,j}= ., the room is not locked and can be freely entered.
Takahashi is currently at the room where A_{i,j}= S, which can also be freely entered.

Each room in the 1-st row, 1-st column, H-th row or W-th column, has an exit.
Each of the other rooms (i,j) is connected to four rooms: (i-1,j), (i+1,j), (i,j-1) and (i,j+1).

Takahashi will use his magic to get out of the building. In one cast, he can do the following:

Move to an adjacent room at most K times, possibly zero. Here, locked rooms cannot be entered.

Then, select and unlock at most K locked rooms, possibly zero. Those rooms will remain unlocked from then on.

His objective is to reach a room with an exit. Find the minimum necessary number of casts to do so.

It is guaranteed that Takahashi is initially at a room without an exit.

Constraints

3 ≤ H ≤ 800

3 ≤ W ≤ 800

1 ≤ K ≤ H×W

Each A_{i,j} is # , . or S.

There uniquely exists (i,j) such that A_{i,j}= S, and it satisfies 2 ≤ i ≤ H-1 and 2 ≤ j ≤ W-1.

Input

Input is given from Standard Input in the following format:

H W K
A_{1,1}A_{1,2}...A_{1,W}
:
A_{H,1}A_{H,2}...A_{H,W}

Output

Print the minimum necessary number of casts.

Sample Input 1

3 3 3
#.#
#S.
###

Sample Output 1

1

Takahashi can move to room (1,2) in one cast.

Sample Input 2

3 3 3
###
#S#
###

Sample Output 2

2

Takahashi cannot move in the first cast, but can unlock room (1,2).
Then, he can move to room (1,2) in the next cast, achieving the objective in two casts.

Sample Input 3

7 7 2
#######
#######
##...##
###S###
##.#.##
###.###
#######

Sample Output 3

2

Solution in C# (Mono 4.6.2.0) (judge verdict: Wrong Answer)

using System;
using System.Collections;
using System.Collections.Generic;
using System.Linq;
using System.Text;

class TEST{
	static void Main(){
		Sol mySol =new Sol();
		mySol.Solve();
	}
}

class Sol{
	public void Solve(){
		
		long Inf = (long) 1e18;
		long[][] min = new long[H][];
		bool[][] used = new bool[H][];
		for(int i=0;i<H;i++){
			min[i] = new long[W];
			used[i] = new bool[W];
			for(int j=0;j<W;j++) min[i][j] = Inf;
		}
		int[][] m = new int[H][];
		for(int i=0;i<H;i++){
			m[i] = new int[W];
			for(int j=0;j<W;j++) m[i][j] = S[i][j] == '#' ? 1 : 0;
		}
		Init(m);
		
		
		int r0 = 0, c0 = 0;
		for(int i=0;i<H;i++){
			for(int j=0;j<W;j++){
				if(S[i][j] == 'S'){
					r0 = i; c0 = j;
				}
			}
		}
		
		var Q = new Queue<int>();
		min[r0][c0] = 0;
		Q.Enqueue(enc(r0,c0));
		while(Q.Count>0){
			var rc = Q.Dequeue();
			int r = decR(rc);
			int c = decC(rc);
			for(int t=0;t<4;t++){
				int nr = r + dy[t];
				int nc = c + dx[t];
				if(!InRange(nr,0,H) || !InRange(nc,0,W))continue;
				if(S[nr][nc] == '#') continue;
				if(min[nr][nc] != Inf) continue;
				min[nr][nc] = min[r][c] + 1;
				Q.Enqueue(enc(nr,nc));
			}
		}
		long ans = Inf;
		
		
		
		
		for(int i=0;i<H;i++){
			for(int j=0;j<W;j++){
				if(min[i][j] > K) continue;
				if(i == 0 || i == H-1 || j == 0 || j == W-1 && min[i][j] < K){
					Console.WriteLine(1);
					return;
				}
				
				long turn = 0;
				turn = (i+1 - 1)/K + 1 + 1; ans = Math.Min(ans,turn);
				turn = (H-i - 1)/K + 1 + 1; ans = Math.Min(ans,turn);
				turn = (j+1 - 1)/K + 1 + 1; ans = Math.Min(ans,turn);
				turn = (W-j - 1)/K + 1 + 1; ans = Math.Min(ans,turn);
				
			}
		}
		
		
		Console.WriteLine(ans);
	}
	
	int[] dy = new int[] {0, 1, 0, -1};
	int[] dx = new int[] {1, 0, -1, 0};
	bool InRange(int t, int l, int r){
		return l <= t && t < r;
	}
	
	void Init(int[][] M){
		acc = new int[H+1][];
		for(int i=0;i<=H;i++){
			acc[i] = new int[W+1];
			if(i == 0) continue;
			for(int j=1;j<=W;j++){
				acc[i][j] = M[i-1][j-1];
			}
		}
		
		for(int i=1;i<=H;i++){
			for(int j=1;j<=W;j++){
				acc[i][j] += acc[i][j-1];
			}
		}
		for(int j=1;j<=W;j++){
			for(int i=1;i<=H;i++){
				acc[i][j] += acc[i-1][j];
			}
		}
	}
	
	static int[][] acc;
	static int cal(int x1,int y1,int x2,int y2){
		//return num of land [x1,x2)*[y1,y2); left-inclusive / right-noninclusive;
		if(x2<x1||y2<y1)return 0;
		return acc[y2][x2]-acc[y2][x1]-acc[y1][x2]+acc[y1][x1];
	}
	
	
	
	
	
	int enc(int r, int c){ return (r*1024) | c;}
	int decR(int rc){ return rc / 1024;}
	int decC(int rc){ return rc % 1024;}
	
	
	int H,W;
	long K;
	String[] S;
	public Sol(){
		var d = ria();
		H = d[0]; W = d[1]; K = d[2];
		S = new String[H];
		for(int i=0;i<H;i++) S[i] = rs();
		
		
	}

	static String rs(){return Console.ReadLine();}
	static int ri(){return int.Parse(Console.ReadLine());}
	static long rl(){return long.Parse(Console.ReadLine());}
	static double rd(){return double.Parse(Console.ReadLine());}
	static String[] rsa(char sep=' '){return Console.ReadLine().Split(sep);}
	static int[] ria(char sep=' '){return Array.ConvertAll(Console.ReadLine().Split(sep),e=>int.Parse(e));}
	static long[] rla(char sep=' '){return Array.ConvertAll(Console.ReadLine().Split(sep),e=>long.Parse(e));}
	static double[] rda(char sep=' '){return Array.ConvertAll(Console.ReadLine().Split(sep),e=>double.Parse(e));}
}