BG-MO 2020 Problem 1

On the sides of $\triangle{ABC}$ points $P,Q \in{AB}$ ( $P$ is between $A$ and $Q$ ) and $R\in{BC}$ are chosen. The points $M$ and $N$ are defined as the intersection point of $AR$ with the segments $CP$ and $CQ$ , respectively. If $BC=BQ$ , $CP=AP$ , $CR=CN$ and $\angle{BPC}=\angle{CRA}$ , prove that $MP+NQ=BR$ .