<p>In a gold mine <code>grid</code> of size <code>m x n</code>, each cell in this mine has an integer representing the amount of gold in that cell, <code>0</code> if it is empty.</p>

<p>Return the maximum amount of gold you can collect under the conditions:</p>

<ul>
	<li>Every time you are located in a cell you will collect all the gold in that cell.</li>
	<li>From your position, you can walk one step to the left, right, up, or down.</li>
	<li>You can&#39;t visit the same cell more than once.</li>
	<li>Never visit a cell with <code>0</code> gold.</li>
	<li>You can start and stop collecting gold from <strong>any </strong>position in the grid that has some gold.</li>
</ul>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<pre>
<strong>Input:</strong> grid = [[0,6,0],[5,8,7],[0,9,0]]
<strong>Output:</strong> 24
<strong>Explanation:</strong>
[[0,6,0],
 [5,8,7],
 [0,9,0]]
Path to get the maximum gold, 9 -&gt; 8 -&gt; 7.
</pre>

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

<pre>
<strong>Input:</strong> grid = [[1,0,7],[2,0,6],[3,4,5],[0,3,0],[9,0,20]]
<strong>Output:</strong> 28
<strong>Explanation:</strong>
[[1,0,7],
 [2,0,6],
 [3,4,5],
 [0,3,0],
 [9,0,20]]
Path to get the maximum gold, 1 -&gt; 2 -&gt; 3 -&gt; 4 -&gt; 5 -&gt; 6 -&gt; 7.
</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;= 15</code></li>
	<li><code>0 &lt;= grid[i][j] &lt;= 100</code></li>
	<li>There are at most <strong>25 </strong>cells containing gold.</li>
</ul>

class Solution:
    def __init__(self):
        self.nodes_visited = set()
        self.max_gold = 0

    def getMaximumGold(self, grid: List[List[int]]) -> int:
        for y_idx in range(len(grid)):
            for x_idx in range(len(grid[0])):
                self.traverse_dfs(x_idx, y_idx, grid, 0)
        return self.max_gold

    def traverse_dfs(self, x_idx, y_idx, grid, running_gold):
        # Boundary conditions.
        if y_idx >= len(grid) or x_idx >= len(grid[0]) or y_idx < 0 or x_idx < 0:
            return
        idx = (x_idx, y_idx)
        val = grid[y_idx][x_idx]
        if idx in self.nodes_visited or val == 0:
            return

        # Iteration.
        self.nodes_visited.add(idx)
        running_gold += val  # Accumulate the gold along the path
        self.max_gold = max(self.max_gold, running_gold)
        self.traverse_dfs(x_idx-1, y_idx, grid, running_gold)
        self.traverse_dfs(x_idx+1, y_idx, grid, running_gold)
        self.traverse_dfs(x_idx, y_idx-1, grid, running_gold)
        self.traverse_dfs(x_idx, y_idx+1, grid, running_gold)
        self.nodes_visited.remove(idx)

16400