Inside acute $\angle{XOY},$ points $M$ and $N$ are taken so that $\angle{XON}=\angle{YOM}$. Point $Q$ is taken on segment $OX$ such that $\angle{NQO}=\angle{MQX}.$ Point $P$ is taken such that $\angle{NPO}=\angle{MPY}.$ Prove the lengths of the broken lines $MPN$ and $MQN$ are equal.