<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:
        m, n = len(grid), len(grid[0])
        directions = [[-1, 0], [0, 1], [1, 0], [0, -1]]

        def dfs(r, c):
            grid[r][c] = -island
            for dr, dc in directions:
                nr = r + dr
                nc = c + dc

                if 0 <= nr < m and 0 <= nc < n and grid[nr][nc] == 1:
                    dfs(nr, nc)            

        island = 0
        for r in range(m):
            for c in range(n):
                if grid[r][c] == 1:
                    island += 1
                    dfs(r, c)    
        counts = defaultdict(int)
        for r in range(m):
            for c in range(n):
                if grid[r][c] != 0:
                    counts[grid[r][c]] += 1

        if len(counts) == 0: return 1

        maxV = max(counts.values())
        for r in range(m):
            for c in range(n):
                if grid[r][c] == 0:
                    island_ids = []
                    for dr, dc in directions:
                        nr, nc = r + dr, c + dc
                        if 0 <= nr < m and 0 <= nc < n and grid[nr][nc] != 0 and grid[nr][nc] not in island_ids:
                            island_ids.append(grid[nr][nc])
                    values = [counts[i] for i in island_ids]
                    maxV = max(maxV, sum(values) + 1)   
        return maxV
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