<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 {
public:
    int largestIsland(vector<vector<int>>& grid) {
        int n = grid.size();
        vector<vector<int>> directions = {{0, 1}, {1, 0}, {0, -1}, {-1, 0}};
        vector<int> area; 
        int islandID = 2; 
        
        function<int(int, int)> dfs = [&](int x, int y) {
            if (x < 0 || y < 0 || x >= n || y >= n || grid[x][y] != 1) return 0;
            grid[x][y] = islandID; 
            int size = 1;
            for (auto dir : directions) {
                size += dfs(x + dir[0], y + dir[1]);
            }
            return size;
        };
        
       
        for (int i = 0; i < n; ++i) {
            for (int j = 0; j < n; ++j) {
                if (grid[i][j] == 1) {
                    area.push_back(dfs(i, j)); 
                    islandID++;
                }
            }
        }

        
        if (area.empty()) return 1;

        
        int result = *max_element(area.begin(), area.end());
        
       
        for (int i = 0; i < n; ++i) {
            for (int j = 0; j < n; ++j) {
                if (grid[i][j] == 0) {
                    unordered_set<int> neighborIslands;
                    int newSize = 1; 
                    
                    for (auto dir : directions) {
                        int ni = i + dir[0], nj = j + dir[1];
                        if (ni >= 0 && nj >= 0 && ni < n && nj < n && grid[ni][nj] > 1) {
                            neighborIslands.insert(grid[ni][nj]);
                        }
                    }
                    
                    for (int id : neighborIslands) {
                        newSize += area[id - 2]; 
                    }
                    
                    result = max(result, newSize);
                }
            }
        }
        
        return result;
    }
};

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