<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:
    vector<vector<int>> dirs = {{0,1}, {1,0}, {0,-1}, {-1,0}};

    int largestIsland(vector<vector<int>>& grid) {
        int n = grid.size();
        vector<int> groupSize; // Stores the size of each group
        groupSize.push_back(0); // No group 0
        groupSize.push_back(0); // No group 1, this will not be used
        int group = 2; // Start group numbering from 2

        // DFS to assign group numbers and calculate their sizes
        for (int i = 0; i < n; ++i) {
            for (int j = 0; j < n; ++j) {
                if (grid[i][j] == 1) {
                    int size = dfs(grid, i, j, group);
                    groupSize.push_back(size);
                    group++;
                }
            }
        }

        // Calculate the largest possible island size without flipping any 0 to 1
        int maxIslandSize = 0;
        for (int size : groupSize) {
            maxIslandSize = max(maxIslandSize, size);
        }

        // Check if flipping a 0 to 1 can create a larger island
        for (int i = 0; i < n; ++i) {
            for (int j = 0; j < n; ++j) {
                if (grid[i][j] == 0) {
                    int possibleSize = 1; // Start with the flipped 0
                    int visitedGroups[4] = {-1, -1, -1, -1}; // Track visited groups for the 4 adjacent cells
                    int index = 0;

                    // Check all 4 adjacent cells
                    for (auto dir : dirs) {
                        int x = i + dir[0];
                        int y = j + dir[1];
                        if (x >= 0 && x < n && y >= 0 && y < n && grid[x][y] > 1) {
                            bool alreadyVisited = false;
                            for (int k = 0; k < index; ++k) {
                                if (visitedGroups[k] == grid[x][y]) {
                                    alreadyVisited = true;
                                    break;
                                }
                            }
                            if (!alreadyVisited) {
                                possibleSize += groupSize[grid[x][y]];
                                visitedGroups[index++] = grid[x][y];
                            }
                        }
                    }
                    maxIslandSize = max(maxIslandSize, possibleSize);
                }
            }
        }

        return maxIslandSize;
    }

private:
    int dfs(vector<vector<int>>& grid, int x, int y, int group) {
        grid[x][y] = group;
        int n = grid.size();
        int size = 1; // Start with the current cell

        // Explore all 4 directions
        for (auto dir : dirs) {
            int i = x + dir[0];
            int j = y + dir[1];
            if (i >= 0 && i < n && j >= 0 && j < n && grid[i][j] == 1) {
                size += dfs(grid, i, j, group);
            }
        }
        return size;
    }
};

287915