The incenter $O$ of an isosceles triangle $ABC$ with $AB=AC$ meets $BC$, $CA$, $AB$ at $K$, $L$, $M$ respectively. Let $N$ be the intersection of lines $OL$ and $KM$ and let $Q$ be the intersection of lines $BN$ and $CA$. Let $P$ be the foot of the perpendicular from $A$ to $BQ$. If we assume that $BP=AP+2PQ$, what are the possible values of $\frac{AB}{BC}$?