RMO 2013 M6

Let $P(x)=x^3+ax^2+b$ and $Q(x)=x^3+bx+a$ , where $a$ and $b$ are nonzero 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 $Q(2013!+1)$ .