Count Tilings with Dominoes and L-Shaped Trominoes

### Problem Tile a 2-by-n board completely using 2-by-1 dominoes and L-shaped three-cell trominoes, allowing rotations. Tiles may not overlap or extend outside the board. Count distinct tile placements modulo 1000000007; placements differ when some adjacent cell pair belongs to one tile in exactly one placement. ```cpp class Solution { public: int numTilings(int n); }; ``` ### Examples **Example 1:** ```text n = 2 Output: 2 Explanation: Use two vertical dominoes or two horizontal dominoes. ``` **Example 2:** ```text n = 4 Output: 11 Explanation: With full-board counts F1=1,F2=2,F3=5 and one-corner-missing count G3=2, F4=F3+F2+2*G3=11. ``` ### Constraints - Board length n lies in [1,1000]. - Return the count modulo 1000000007. ### Contract considerations Preserve rotation and the original notion of distinct tile placements. ```hint Read the boundary condition Which incomplete boundary shape is needed beside a full-board DP state? ``` ### Boundary reminders Preserve rotation and the original notion of distinct tile placements.