<p>You are given an <code>n x n</code> binary matrix <code>grid</code>. You are allowed to change <strong>at most one</strong> <code>0</code> to be <code>1</code>.</p>

<p>Return <em>the size of the largest <strong>island</strong> in</em> <code>grid</code> <em>after applying this operation</em>.</p>

<p>An <strong>island</strong> is a 4-directionally connected group of <code>1</code>s.</p>

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

<pre>
<strong>Input:</strong> grid = [[1,0],[0,1]]
<strong>Output:</strong> 3
<strong>Explanation:</strong> Change one 0 to 1 and connect two 1s, then we get an island with area = 3.
</pre>

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

<pre>
<strong>Input:</strong> grid = [[1,1],[1,0]]
<strong>Output:</strong> 4
<strong>Explanation: </strong>Change the 0 to 1 and make the island bigger, only one island with area = 4.</pre>

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

<pre>
<strong>Input:</strong> grid = [[1,1],[1,1]]
<strong>Output:</strong> 4
<strong>Explanation:</strong> Can&#39;t change any 0 to 1, only one island with area = 4.
</pre>

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

<ul>
	<li><code>n == grid.length</code></li>
	<li><code>n == grid[i].length</code></li>
	<li><code>1 &lt;= n &lt;= 500</code></li>
	<li><code>grid[i][j]</code> is either <code>0</code> or <code>1</code>.</li>
</ul>

class Solution:
    def largestIsland(self, grid: List[List[int]]) -> int:
        def dfs(i, j, fill):
            if i < 0 or i >= m or j < 0 or j >= n or grid[i][j]!= 1:
                return 0
            grid[i][j] = fill
            return 1 + sum(dfs(i + dx, j + dy, fill) for dx, dy in dirs)

        dirs = ((-1, 0), (1, 0), (0, -1), (0, 1))
        m, n = len(grid), len(grid[0])
        res = 0
        area = {}
        # Color each island into 2, 3, 4, ...
        idx = 2
        for i in range(m):
            for j in range(n):
                if grid[i][j] == 1:
                    area[idx] = dfs(i, j, idx)
                    res = max(res, area[idx])
                    idx += 1
        for i in range(m):
            for j in range(n):
                if not grid[i][j]:
                    idxs = set(grid[i+dx][j+dy] for dx, dy in dirs if i + dx >= 0 and i + dx < m and j + dy >= 0 and j + dy < n and grid[i+dx][j+dy])
                    res = max(res, 1 + sum(area[idx] for idx in idxs))
        return res
    
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