For some real number $a$, define two parabolas on the coordinate plane with equations $x = y^2 + a$ and $y = x^2 + a$. Suppose there are $3$ lines, each tangent to both parabolas, that form an equilateral triangle with positive area $s$. If $s^2 = \frac{p}{q}$ for coprime positive integers $p$, $q$, find $p + q$.

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