Easy but nice

[ins-]

Let $ABCD$ is a convex quadrilateral. $k_1$ is circumcircle of the triangle $BCD$. $M$ and $N$ are the intersection points of $k_1$ with the segments $AB$ and $AD$, respectively. $k_2$ is the circumcircle of the triangle $AMN$. $P$ is the intersection point of the diagonals and $Q$ is the intersection point of the segment $AP$ with $k_2$. Prove that the points $B$, $M$, $P$, $Q$ are concyclic.

[ins-]

Let $ABCD$ is a convex quadrilateral. $k_1$ is circumcircle of the triangle $BCD$. $M$ and $N$ are the intersection points of $k_1$ with the segments $AB$ and $AD$, respectively. $k_2$ is the circumcircle of the triangle $AMN$. $P$ is the intersection point of the diagonals and $Q$ is the intersection point of the segment $AP$ with $k_2$. $K$ is the intersection point of $k_2$ and the circumcircle of the triangle $BDQ$. $L$ is the intersection point of $AK$ and the diagonal $BD$. Prove that the points $K$, $L$, $P$, $Q$ and the points $M$, $N$, $P$, $L$ are concyclic.

It is a harder problem, related to the problem above.