<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:
        island_index = 2
        n, m = len(grid), len(grid[0])

        def neighbors(i, j):
            for n_i, n_j in ((i + 1, j), (i - 1, j), (i, j + 1), (i, j - 1)):
                if 0 <= n_i < n and 0 <= n_j < m:
                    yield n_i, n_j

        def dfs(i, j):
            grid[i][j] = island_index
            return 1 + sum(
                dfs(n_i, n_j) for n_i, n_j in neighbors(i, j) if grid[n_i][n_j] == 1
            )

        island_areas = {0: 0}
        max_area = 0
        for i in range(n):
            for j in range(m):
                if grid[i][j] == 1:
                    area = dfs(i, j)
                    max_area = max(area, max_area)
                    island_areas[island_index] = area
                    island_index += 1

        for i in range(n):
            for j in range(m):
                if grid[i][j] == 0:
                    near_islands = set(grid[n_i][n_j] for n_i, n_j in neighbors(i, j))
                    area = 1 + sum(island_areas[index] for index in near_islands)
                    max_area = max(area, max_area)
        return max_area

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