5.2.4.32 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 about B a trapezoid \( A B C D \) with bases \( A D \) and \( B C \) such that \( A D: B C=5: 3 \), the diagonals intersect at the point \( M \). Express the vectors \( \overrightarrow{M A}, \overrightarrow{M B}, \overrightarrow{M C}, \overrightarrow{M D} \) in terms of the vectors \( \vec{a}=\overrightarrow{A B} \) and \( \vec{b}=\overrightarrow{B C} \)