Problem Analysis #6

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.

A point starts at $(2,1)$. Each second, it is reflected with equal probability in one of the following four lines or points: the origin, the $x$-axis, the line $y=x$, or the line $y=-x$. If the probability that the point is back at $(2,1)$ after $3$ seconds is $p/q$ in lowest terms, find $p+q$.