5.2.4.31 Various problems on the plane

condition: In the triangle \( A B C \) the points \( M, N, K \) are located respectively on the sides \( A B, B C, A C \) so that \( A M: M B=1: 2 \), \( C N: N B=1: 3, A K=K C \). The segments \( M N \) and \( B K \) intersect at the point \( P \). Find the ratio of \( M P: P N \) and \( B P: P K \).