An equiangular hexagon has side lengths $1, 1, a, 1, 1, a$ in that order. Given that there exists a circle that intersects the hexagon at $12$ distinct points, we have $M < a < N$ for some real numbers $M$ and $N$. Determine the minimum possible value of the ratio $\frac{N}{M}$.

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