Let $$ABCD$$ be a unit square. Suppose $$M$$ and $$N$$ are points on $$BC$$ and $$CD$$ respectively such that the perimeter of triangle $$MCN$$ is $$2$$. Let $$O$$ be the circumcentre of triangle $$MAN$$, and $$P$$ be the circumcentre of triangle $$MON$$. If $$(\frac{OP}{OA})^{2}=\frac{m}{n}$$ for some relatively prime positive integers $$m$$ and $$n$$, find the value of $$m+n$$.