Problem Analysis #4

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.

In three-dimensional space, take $A(0,1,1)$ and $O(0,0,0)$. Let $P$ be a point on the $xy$-plane satisfying

Let $L$ be the total arc length of the set of all possible points $P$. If $L=p\pi/q$ in lowest terms, find $p+q$.