Given $\triangle ABC$ with $\angle A = 15^{\circ}$, let $M$ be the midpoint of $BC$ and let $E$ and $F$ be points on ray $BA$ and $CA$ respectively such that $BE = BM = CF$. Let $R_1$ be the radius of $(MEF)$ and $R_2$ be the radius of $(AEF)$. If $\frac{R_1^2}{R_2^2} = a - \sqrt{b + \sqrt{c}}$ where $a, b, c$ are integers, find $a^{b^{c}}$.

Let's think step by step and output the final answer within \boxed{}.
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