Problem Analysis #10

Return a single integer only. The problem has been revised so that no final answer is an irrational number, a parameterized expression, a range, a proof, a drawing, a graph, a locus equation, or a region description.

Let $a>0$, and consider the parabola $y=x^2$. Starting from $P(a,a^2)$, let $Q\ne P$ be the other intersection of the normal line to the parabola at $P$ with the parabola. Let $R\ne Q$ be the other intersection of the normal line at $Q$ with the parabola.

If the minimum possible $x$-coordinate of $R$ is $M$, write $M^2=p/q$ in lowest terms and find $p+q$.