<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:
        visited = set()

        def dfs(m: int, n: int) -> int:
            if (m, n) not in visited and 0 <= m < len(grid) and 0 <= n < len(grid[0]):
                visited.add((m, n))

                if grid[m][n] == 1:
                    island_indices.append((m, n))
                    return 1 + dfs(m-1, n) + dfs(m, n-1) + dfs(m+1, n) + dfs(m, n+1)

            return 0

        answer = 0
        island_id = 1
        island_map = {}

        for m in range(len(grid)):
            for n in range(len(grid[0])):
                if (m, n) not in visited and grid[m][n] == 1:
                    island_indices = []
                    answer = max(answer, dfs(m, n))
                    island_map[island_id] = island_indices
                    island_id += 1
        
        answer = 0
        for idx, indices in island_map.items():
            for x, y in indices:
                grid[x][y] = idx
                answer = max(answer, len(indices))

        for m in range(len(grid)):
            for n in range(len(grid[0])):
                if grid[m][n] == 0:
                    surr_islands = set()
                    if m > 0:
                        surr_islands.add(grid[m-1][n])
                    if n > 0:
                        surr_islands.add(grid[m][n-1])
                    if m+1 < len(grid):
                        surr_islands.add(grid[m+1][n])
                    if n+1 < len(grid[0]):
                        surr_islands.add(grid[m][n+1])

                    tmp_answer = 1
                    for isl in surr_islands:
                        tmp_answer += len(island_map.get(isl, []))
                    answer = max(answer, tmp_answer)
                
        return answer
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