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.

[asy]
pair h = (1, 0), v = (0, 1), o = (0, 0);
for(int i = 1; i < 5; ++i)
{
  o = (i*i/2 + i, 0);
  draw(o -- o + i*v -- o + i*v + h -- o + h + v -- o + i*h + v -- o + i*h -- cycle);
  string s = "$L_" + (string)(i) + "$";
  label(s, o + ((i / 2), -1));
  for(int j = 1; j < i; ++j)
  {
    draw(o + j*v -- o + j*v + h);
    draw(o + j*h -- o + j*h + v);
  }
}
label("...", (18, 0.5));
[/asy]

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.