<p>You are given an <code>m x n</code> <code>grid</code>. Each cell of <code>grid</code> represents a street. The street of <code>grid[i][j]</code> can be:</p>

<ul>
	<li><code>1</code> which means a street connecting the left cell and the right cell.</li>
	<li><code>2</code> which means a street connecting the upper cell and the lower cell.</li>
	<li><code>3</code> which means a street connecting the left cell and the lower cell.</li>
	<li><code>4</code> which means a street connecting the right cell and the lower cell.</li>
	<li><code>5</code> which means a street connecting the left cell and the upper cell.</li>
	<li><code>6</code> which means a street connecting the right cell and the upper cell.</li>
</ul>
<img alt="" src="https://assets.leetcode.com/uploads/2020/03/05/main.png" style="width: 450px; height: 708px;" />
<p>You will initially start at the street of the upper-left cell <code>(0, 0)</code>. A valid path in the grid is a path that starts from the upper left cell <code>(0, 0)</code> and ends at the bottom-right cell <code>(m - 1, n - 1)</code>. <strong>The path should only follow the streets</strong>.</p>

<p><strong>Notice</strong> that you are <strong>not allowed</strong> to change any street.</p>

<p>Return <code>true</code><em> if there is a valid path in the grid or </em><code>false</code><em> otherwise</em>.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2020/03/05/e1.png" style="width: 455px; height: 311px;" />
<pre>
<strong>Input:</strong> grid = [[2,4,3],[6,5,2]]
<strong>Output:</strong> true
<strong>Explanation:</strong> As shown you can start at cell (0, 0) and visit all the cells of the grid to reach (m - 1, n - 1).
</pre>

<p><strong class="example">Example 2:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2020/03/05/e2.png" style="width: 455px; height: 293px;" />
<pre>
<strong>Input:</strong> grid = [[1,2,1],[1,2,1]]
<strong>Output:</strong> false
<strong>Explanation:</strong> As shown you the street at cell (0, 0) is not connected with any street of any other cell and you will get stuck at cell (0, 0)
</pre>

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

<pre>
<strong>Input:</strong> grid = [[1,1,2]]
<strong>Output:</strong> false
<strong>Explanation:</strong> You will get stuck at cell (0, 1) and you cannot reach cell (0, 2).
</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;= 300</code></li>
	<li><code>1 &lt;= grid[i][j] &lt;= 6</code></li>
</ul>

DIR_OPS = {
    "N": "S",
    "S": "N",
    "W": "E",
    "E": "W"
}


class Grid:
    def __init__(self, grid, next_output):
        self.x = 0
        self.y = 0
        self._grid = grid
        self.m = len(self._grid[0])
        self.n = len(self._grid)

        # N = 0, E = 1, S = 2, W = 3
        self._input_output_map = {
            1: ["W", "E"],
            2: ["N", "S"],
            3: ["W", "S"],
            4: ["S", "E"],
            5: ["W", "N"],
            6: ["N", "E"],
        }
        self._visited = set()
    
    def _descend(self, next_output):
        curr = self._grid[self.y][self.x]
        x = self.x
        y = self.y

        #print("Currently", (x, y), curr)
        if (x, y) == (self.m - 1, self.n - 1):
            return (x, y)

        if (x, y) in self._visited:
            #print("already visited", (x, y))
            return (x, y)
        self._visited.add((x, y))

        directions = set(self._input_output_map[curr])

        if next_output not in directions:
            #print("not in directions", (x, y), next_output, directions)
            return (x, y)

        came_from = next_output
        input = (directions - {next_output}).pop()
        next_output = DIR_OPS[input]

        if input == "N":
            y -=1 
        elif input == "S":
            y += 1
        elif input == "W":
            x -=1 
        elif input == "E":
            x += 1
        
        #print("moving", input)
        if y >= len(self._grid) or x >= len(self._grid[0]):
            #print("out of bounds", (self.x , self.y))
            return (self.x, self.y)
        
        if y < 0 or x < 0:
            return (self.x, self.y)
        
        curr = self._grid[y][x]
        directions = set(self._input_output_map[curr])
        if next_output not in directions:
            return (self.x, self.y)

        #print("came from", came_from, "moving", input, "to", curr, self._grid, x, y)

        self.x, self.y = x, y
        return self._descend(next_output)
    
    def descend(self):
        curr = self._grid[self.y][self.x]

        for next_output in self._input_output_map[curr]:
            #print("\n\n----start next_output is", next_output)
            self.x, self.y = 0, 0
            self._visited = set()
            last = self._descend(next_output)


            if last == (self.m - 1, self.n - 1):
                return True
        return False

        
class Solution:
    def hasValidPath(self, grid: List[List[int]]) -> bool:
        
        g = Grid(grid, "S")
        return g.descend()
        
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