<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:
        zeroes = []

        m = len(grid)
        n = len(grid)
        directions = [[1, 0], [-1, 0], [0, 1], [0, -1]]
        islandMap = defaultdict(int)
        sumMax = 0

        def areaize(row, col, count):
            nonlocal directions
            nonlocal m
            nonlocal n
            nonlocal islandMap
            seen = set()
            nonlocal sumMax
            
            q = deque()

            q.append((row, col))
            seen.add((row, col))
            while q:
                row, col = q.popleft()

                for dx, dy in directions:
                    newRow = row + dx
                    newCol = col + dy

                    if 0 <= newRow < m and 0 <= newCol < n and (newRow, newCol) not in seen and grid[newRow][newCol] == 1:
                        seen.add((newRow, newCol))
                        q.append((newRow, newCol))

            for row, col in seen:
                grid[row][col] = count
            
            islandMap[count] = len(seen)
            sumMax = max(sumMax, len(seen))

        count = 2
        for row in range(m):
            for col in range(n):
                if grid[row][col] == 0:
                    zeroes.append([row, col])
                elif grid[row][col] == 1:
                    areaize(row, col, count)
                    count += 1
        for row, col in zeroes:
            total = 1
            seen = set()
            for dx, dy in directions:
                newRow = row + dx
                newCol = col + dy

                if 0 <= newRow < m and 0 <= newCol < n and grid[newRow][newCol] not in seen:
                    seen.add(grid[newRow][newCol])
                    total += islandMap[grid[newRow][newCol]]
            
            sumMax = max(total, sumMax)
                    
        return sumMax

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