Let $ABCD$ be a cyclic quadrilateral with circumcenter $O.$ Diagonals $AC$ and $BD$ meet at $E.$ $F$ and $G$ are points on segments $AB$ and $CD$ respectively. Suppose $AF=15,$ $FB=13,$ $BE=30,$ $ED=13,$ $DG=7.5,$ and $GC=6.5.$ Let $P$ be a point such that $PF\perp AB$ and $PG\perp CD.$ Find $\frac{PE}{PO}.$