Question 14
Let $A_1A_2A_3A_4A_5A_6A_7A_8$ be a regular octagon. Let $M_1,$ $M_3,$ $M_5,$ and $M_7$ be the midpoints of sides $\overline{A_1A_2},$ $\overline{A_3A_4},$ $\overline{A_5A_6},$ and $\overline{A_7A_8},$ respectively. For $i = 1,$ $3,$ $5,$ $7,$ ray $R_i$ is constructed from $M_i$ towards the interior of the octagon such that $R_1 \perp R_3,$ $R_3 \perp R_5,$ $R_5 \perp R_7,$ and $R_7 \perp R_1.$ Pairs of rays $R_1$ and $R_3,$ $R_3$ and $R_5,$ $R_5$ and $R_7,$ and $R_7$ and $R_1$ meet at $B_1,$ $B_3,$ $B_5,$ and $B_7,$ respectively. If $B_1B_3 = A_1A_2,$ then $\cos 2\angle A_3M_3B_1$ can be written in the form $m - \sqrt{n},$ where $m$ and $n$ are positive integers. Find $m + n.$