<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:
        rows = len(grid)
        cols = len(grid[0])

        zeros = []
        island_sz = defaultdict(int)
        visited = set()

        dirs = [(0, 1), (0, -1), (1, 0), (-1, 0)]
        island_idx = 2

        for i in range(rows):
            for j in range(cols):
                if (i, j) in visited:
                    continue

                if grid[i][j] == 0:
                    visited.add((i,j))
                    zeros.append((i, j))
                    continue
                
                bfs = deque()
                bfs.append((i,j))
                while bfs:
                    x, y = bfs.popleft()
                    if (x, y) in visited:
                        continue
                    for d in dirs:
                        dx = x + d[0]
                        dy = y + d[1]
                        if 0 <= dx < rows and 0 <= dy < cols and (dx, dy) not in visited and grid[dx][dy] == 1:
                            bfs.append((dx, dy))
                    
                    visited.add((x, y))
                    island_sz[island_idx] += 1
                    grid[x][y] = island_idx
                 
                island_idx += 1

        if not zeros:
            return island_sz[2]
        if not island_sz:
            return 1

        max_size = 0
        for i, j in zeros:
            curr = 0
            island_visit = set()
            for d in dirs:
                di = i + d[0]
                dj = j + d[1]
                if 0 <= di < rows and 0 <= dj < cols and (di, dj) != 0 and grid[di][dj] not in island_visit:
                    curr += island_sz[grid[di][dj]]
                    island_visit.add(grid[di][dj])
            max_size = max(max_size, curr + 1)
        
        return max_size
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