Let $P(x) = x^3 + ax^2 + b$ and $Q(x) = x^3 + bx + a$, where $a,b$ are non-zero real numbers. Suppose that the roots of the equation $P(x) = 0$ are the reciprocals of the roots of the equation $Q(x) = 0.$ Prove that $a$ and $b$ are integers. Find the greatest common divisor of $P(2013! + 1)$ and $Q(2013! + 1).$

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