<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])
        self.island_id = -1
        self.island_areas = {}
        self.directions = [(1,0), (-1,0), (0,1), (0,-1)]
        
        for i in range(m):
            for j in range(n):
                if grid[i][j] == 1:
                    island_area = self.dfs(grid, i, j) # find the area
                    self.island_areas[self.island_id] = island_area
                    self.island_id -= 1
        
        max_area = 0

        for i in range(m):
            for j in range(n):
                if grid[i][j] == 0:
                    area = 1
                    surrounding = set()

                    for dx, dy in self.directions:
                        nr, nc = i + dx, j + dy
                        if 0 <= nr < m and 0 <= nc < n and grid[nr][nc] != 0:
                            surrounding.add(grid[nr][nc])
                    
                    for island_id in surrounding:
                        area += self.island_areas[island_id]
                    max_area = max(max_area, area)
        
        return max_area if max_area else len(grid) ** 2
    
    def dfs(self, grid, i, j):
        if 0 <= i < len(grid) and 0 <= j < len(grid[0]) and grid[i][j] == 1:
            grid[i][j] = self.island_id
            return 1 + self.dfs(grid, i + 1, j) + self.dfs(grid, i, j + 1) + self.dfs(grid, i - 1, j) + self.dfs(grid, i, j - 1)
        else:
            return 0
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