Rotate a Stone Box and Settle Its Contents

### Problem `boxGrid` contains stones `#`, fixed obstacles `*`, and empty cells `.`. Rotate this rectangular box ninety degrees clockwise, then let stones fall downward until blocked by an obstacle, a stone, or the bottom. Obstacles rotate with the box but do not fall. Stones are initially supported. Return the matrix with its dimensions exchanged after settling. ```cpp class Solution { public: vector<vector<char>> rotateTheBox(vector<vector<char>>& boxGrid); }; ``` ### Examples **Example 1:** ```text boxGrid = [["#",".","*","#","."]] Output: [["."],["#"],["*"],["."],["#"]] Explanation: Stones settle separately on the two sides of the obstacle. ``` **Example 2:** ```text boxGrid = [["#","#","."],["#","#","."]] Output: [[".","."],["#","#"],["#","#"]] Explanation: Each original row compacts rightward before mapping the rotation. ``` ### Constraints - `m == boxGrid.length` and `n == boxGrid[i].length`. - `1 <= m,n <= 500`. - Cells are `#`, `*`, or `.`. - Initially every stone rests on an obstacle, another stone, or the bottom. ### Clarifying Questions Rotated gravity corresponds to rightward movement in each original row. Stones do not shift horizontally in the rotated coordinate system. ```hint Inspect the structure Which objects move under gravity after the rotation, and what must happen to the obstacles? ``` ### Reasoning checkpoints - Model the specified clockwise rotation and downward settling without moving an obstacle relative to the box. - Preserve stone multiplicity and show that no stone can fall farther in the final grid. ### Optional extensions How would rotating first and then compacting output columns implement the same behavior?