<p>A storekeeper is a game in which the player pushes boxes around in a warehouse trying to get them to target locations.</p>

<p>The game is represented by an <code>m x n</code> grid of characters <code>grid</code> where each element is a wall, floor, or box.</p>

<p>Your task is to move the box <code>&#39;B&#39;</code> to the target position <code>&#39;T&#39;</code> under the following rules:</p>

<ul>
	<li>The character <code>&#39;S&#39;</code> represents the player. The player can move up, down, left, right in <code>grid</code> if it is a floor (empty cell).</li>
	<li>The character <code>&#39;.&#39;</code> represents the floor which means a free cell to walk.</li>
	<li>The character<font face="monospace">&nbsp;</font><code>&#39;#&#39;</code><font face="monospace">&nbsp;</font>represents the wall which means an obstacle (impossible to walk there).</li>
	<li>There is only one box <code>&#39;B&#39;</code> and one target cell <code>&#39;T&#39;</code> in the <code>grid</code>.</li>
	<li>The box can be moved to an adjacent free cell by standing next to the box and then moving in the direction of the box. This is a <strong>push</strong>.</li>
	<li>The player cannot walk through the box.</li>
</ul>

<p>Return <em>the minimum number of <strong>pushes</strong> to move the box to the target</em>. If there is no way to reach the target, return <code>-1</code>.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2019/11/06/sample_1_1620.png" style="width: 500px; height: 335px;" />
<pre>
<strong>Input:</strong> grid = [[&quot;#&quot;,&quot;#&quot;,&quot;#&quot;,&quot;#&quot;,&quot;#&quot;,&quot;#&quot;],
               [&quot;#&quot;,&quot;T&quot;,&quot;#&quot;,&quot;#&quot;,&quot;#&quot;,&quot;#&quot;],
               [&quot;#&quot;,&quot;.&quot;,&quot;.&quot;,&quot;B&quot;,&quot;.&quot;,&quot;#&quot;],
               [&quot;#&quot;,&quot;.&quot;,&quot;#&quot;,&quot;#&quot;,&quot;.&quot;,&quot;#&quot;],
               [&quot;#&quot;,&quot;.&quot;,&quot;.&quot;,&quot;.&quot;,&quot;S&quot;,&quot;#&quot;],
               [&quot;#&quot;,&quot;#&quot;,&quot;#&quot;,&quot;#&quot;,&quot;#&quot;,&quot;#&quot;]]
<strong>Output:</strong> 3
<strong>Explanation:</strong> We return only the number of times the box is pushed.</pre>

<p><strong class="example">Example 2:</strong></p>

<pre>
<strong>Input:</strong> grid = [[&quot;#&quot;,&quot;#&quot;,&quot;#&quot;,&quot;#&quot;,&quot;#&quot;,&quot;#&quot;],
               [&quot;#&quot;,&quot;T&quot;,&quot;#&quot;,&quot;#&quot;,&quot;#&quot;,&quot;#&quot;],
               [&quot;#&quot;,&quot;.&quot;,&quot;.&quot;,&quot;B&quot;,&quot;.&quot;,&quot;#&quot;],
               [&quot;#&quot;,&quot;#&quot;,&quot;#&quot;,&quot;#&quot;,&quot;.&quot;,&quot;#&quot;],
               [&quot;#&quot;,&quot;.&quot;,&quot;.&quot;,&quot;.&quot;,&quot;S&quot;,&quot;#&quot;],
               [&quot;#&quot;,&quot;#&quot;,&quot;#&quot;,&quot;#&quot;,&quot;#&quot;,&quot;#&quot;]]
<strong>Output:</strong> -1
</pre>

<p><strong class="example">Example 3:</strong></p>

<pre>
<strong>Input:</strong> grid = [[&quot;#&quot;,&quot;#&quot;,&quot;#&quot;,&quot;#&quot;,&quot;#&quot;,&quot;#&quot;],
               [&quot;#&quot;,&quot;T&quot;,&quot;.&quot;,&quot;.&quot;,&quot;#&quot;,&quot;#&quot;],
               [&quot;#&quot;,&quot;.&quot;,&quot;#&quot;,&quot;B&quot;,&quot;.&quot;,&quot;#&quot;],
               [&quot;#&quot;,&quot;.&quot;,&quot;.&quot;,&quot;.&quot;,&quot;.&quot;,&quot;#&quot;],
               [&quot;#&quot;,&quot;.&quot;,&quot;.&quot;,&quot;.&quot;,&quot;S&quot;,&quot;#&quot;],
               [&quot;#&quot;,&quot;#&quot;,&quot;#&quot;,&quot;#&quot;,&quot;#&quot;,&quot;#&quot;]]
<strong>Output:</strong> 5
<strong>Explanation:</strong> push the box down, left, left, up and up.
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>m == grid.length</code></li>
	<li><code>n == grid[i].length</code></li>
	<li><code>1 &lt;= m, n &lt;= 20</code></li>
	<li><code>grid</code> contains only characters <code>&#39;.&#39;</code>, <code>&#39;#&#39;</code>, <code>&#39;S&#39;</code>, <code>&#39;T&#39;</code>, or <code>&#39;B&#39;</code>.</li>
	<li>There is only one character <code>&#39;S&#39;</code>, <code>&#39;B&#39;</code>, and <code>&#39;T&#39;</code> in the <code>grid</code>.</li>
</ul>

class Solution:
    def minPushBox(self, grid: List[List[str]]) -> int:
        
        adjs = ((-1, 0), (1, 0), (0, -1), (0, 1))

        '''for row in grid:
            print(row)
        print('')'''

        @cache
        def validpushes(s, b):
            #print(s, b)
            cur = [s]
            newcur = []
            invis = set([s])
            flag = 0
            boxsides = set([(b[0]+adj[0], b[1]+adj[1]) for adj in adjs])
            res = []
            if s in boxsides:
                res.append(s)
            while True:
                if len(cur) == 0:
                    break
                for el in cur:
                    for newadj in adjs:
                        newel = (el[0]+newadj[0], el[1]+newadj[1])
                        if newel != b and 0 <= newel[0] < len(grid) and 0 <= newel[1] < len(grid[0]) and grid[newel[0]][newel[1]] != '#' and newel not in invis:
                            invis.add(newel)
                            newcur.append(newel)
                            if newel in boxsides:
                                res.append(newel)
                cur, newcur = newcur, []
            
            return res
        
        
        vispush = set([])
        t = None
        s = None
        b = None
        for i in range(len(grid)):
            for j in range(len(grid[0])):
                if grid[i][j] == 'T':
                    t = (i, j)
                    grid[i][j] = '.'
                elif grid[i][j] == 'S':
                    s = (i, j)
                    grid[i][j] = '.'
                elif grid[i][j] == 'B':
                    b = (i, j)
                    grid[i][j] = '.'

        '''print(validpushes((3, 3), (3, 2)))
        print(validpushes((4, 2), (3, 2)))'''

        @cache
        def itero(s, b):
            #print(s, b)
            if (s, b) in vispush:
                return +inf
            vispush.add((s, b))
            if b == t:
                return 0
            
            oute = +inf

            valpushes = validpushes(s, b)
            #print(s, b, valpushes)

            for valp in valpushes:
                newboxpos = (2*b[0] - valp[0], 2*b[1] - valp[1])
                if 0 <= newboxpos[0] < len(grid) and 0 <= newboxpos[1] < len(grid[0]) and grid[newboxpos[0]][newboxpos[1]] != '#':
                    
                    oute = min(oute, 1+itero(b, newboxpos))

            return oute
        

        '''path = [
            #s, b
            [(4, 6), (2, 5)],
            [(1, 5), (2, 5)],
            [(3, 6), (3, 5)],
            [(3, 5), (3, 4)],
            [(3, 4), (3, 3)],
            [(4, 2), (3, 2)],
            [(3, 2), (2, 2)],
            [(1, 1), (1, 2)]



        ]

        for node in path[::-1]:
            print(node, itero(node[0], node[1]))
            print(vispush)
            print()'''

        val = itero(s, b)
        if val > 10**10:
            return -1
        return val
18000