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.Generic;
using System.Linq;
using System.Text;
using static System.Console;
using static System.Math;

namespace AtTest._700problems
{
    class AGC014_C
    {
        static void Main(string[] args)
        {
            var sw = new System.IO.StreamWriter(OpenStandardOutput()) { AutoFlush = false };
            SetOut(sw);

            Method(args);

            Out.Flush();
        }

        static void Method(string[] args)
        {
            int[] hwk = ReadInts();
            int h = hwk[0];
            int w = hwk[1];
            int k = hwk[2];
            bool[,] map = new bool[h, w];
            int sX = 0, sY = 0;
            for (int i = 0; i < h; i++)
            {
                string s = Read();
                for (int j = 0; j < w; j++)
                {
                    if (s[j] == '#') continue;
                    if (s[j] == 'S')
                    {
                        sX = j;
                        sY = i;
                    }
                    map[i, j] = true;
                }
            }
            long[,] distances = new long[h, w];
            for (int i = 0; i < h; i++)
            {
                for (int j = 0; j < w; j++)
                {
                    distances[i, j] = long.MaxValue / 4;

                }
            }
            int[] dx = new int[4] { 1, -1, 0, 0 };
            int[] dy = new int[4] { 0, 0, 1, -1 };
            PriorityQueue<int[]> queue = new PriorityQueue<int[]>();
            queue.Enqueue(0, new int[2] { sX, sY });
            while (queue.Count > 0)
            {
                var pair = queue.Dequeue();
                long distance = pair.Key;
                int x = pair.Value[0];
                int y = pair.Value[1];

                if (Compare(distance, distances[y, x], k)) continue;

                distances[y, x] = distance;
                for (int i = 0; i < 4; i++)
                {
                    int nx = x + dx[i];
                    int ny = y + dy[i];
                    if (nx < 0 || w <= nx || ny < 0 || h <= ny)
                    {
                        continue;
                    }

                    long newDistance = distance;
                    if (map[ny, nx])
                    {
                        newDistance+=1000000;
                    }
                    else
                    {
                        newDistance++;
                    }
                    if (Compare(newDistance, distances[ny, nx], k)) continue;
                    queue.Enqueue(newDistance, new int[2] { nx, ny });
                }
            }
            long res = long.MaxValue;
            for (int i = 0; i < h; i++)
            {
                for(int j = 0; j < w; j++)
                {
                    if (i == 0 || i == h - 1 || j == 0 || j == w - 1)
                    {
                        long tmp = GetCnt(distances[i, j], k);
                        res = Min(res, tmp);
                    }
                }
            }
            WriteLine(res);
        }

        static long GetCnt(long distance, int k)
        {
            long moveCnt = (distance / 1000000) / k;
            if ((distance / 1000000) % k > 0) moveCnt++;
            long breakCnt = (distance % 1000000) / k + 1;
            if ((distance % 1000000) % k > 0) breakCnt++;
            return Max(moveCnt, breakCnt);
        }

        static bool Compare(long myDistance,long otherDistance,int k)
        {
            long myCnt = GetCnt(myDistance, k);
            long otherCnt = GetCnt(otherDistance, k);
            return myCnt >= otherCnt;
        }

        class PriorityQueue<T>
        {
            private readonly List<KeyValuePair<long, T>> list;

            public int Count { get; private set; }

            public PriorityQueue()
            {
                list = new List<KeyValuePair<long, T>>();
                Count = 0;
            }

            private void Add(KeyValuePair<long, T> pair)
            {
                if (Count == list.Count)
                {
                    list.Add(pair);
                }
                else
                {
                    list[Count] = pair;
                }
                Count++;
            }

            private void Swap(int a, int b)
            {
                KeyValuePair<long, T> tmp = list[a];
                list[a] = list[b];
                list[b] = tmp;
            }

            public void Enqueue(long key, T value)
            {
                Add(new KeyValuePair<long, T>(key, value));
                int c = Count - 1;
                while (c > 0)
                {
                    int p = (c - 1) / 2;
                    if (list[c].Key >= list[p].Key) break;

                    Swap(p, c);
                    c = p;
                }
            }

            public KeyValuePair<long, T> Top()
            {
                return list[0];
            }

            public KeyValuePair<long, T> Dequeue()
            {
                KeyValuePair<long, T> pair = list[0];
                Count--;
                if (Count == 0) return pair;

                list[0] = list[Count];
                int p = 0;
                while (true)
                {
                    int c1 = p * 2 + 1;
                    int c2 = p * 2 + 2;
                    if (c1 >= Count) break;

                    int c = (c2 >= Count || list[c1].Key < list[c2].Key)
                        ? c1 : c2;
                    if (list[c].Key >= list[p].Key) break;

                    Swap(p, c);
                    p = c;
                }
                return pair;
            }
        }

        private static string Read() { return ReadLine(); }
        private static char[] ReadChars() { return Array.ConvertAll(Read().Split(), a => a[0]); }
        private static int ReadInt() { return int.Parse(Read()); }
        private static long ReadLong() { return long.Parse(Read()); }
        private static double ReadDouble() { return double.Parse(Read()); }
        private static int[] ReadInts() { return Array.ConvertAll(Read().Split(), int.Parse); }
        private static long[] ReadLongs() { return Array.ConvertAll(Read().Split(), long.Parse); }
        private static double[] ReadDoubles() { return Array.ConvertAll(Read().Split(), double.Parse); }
    }
}