OMM 2018 Problem 2

For each positive integer $m$ , we define $L_m$ as the figure that is obtained by overlapping two $1 \times m$ and $m \times 1$ rectangles in such a way that they coincide at the $1 \times 1$ square at their ends, as shown in the figure.

Using some figures $L_{m_1}, L_{m_2}, \dots, L_{m_k}$ , we cover an $n \times n$ board completely, in such a way that the edges of the figure coincide with lines in the board. Among all possible coverings of the board, find the minimal possible value of $m_1 + m_2 + \dots + m_k$ .

Note: In covering the board, the figures may be rotated or reflected, and they may overlap or not be completely contained within the board.