Beautiful concyclic

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Let $ABCD$ is a convex quadrilateral. $k_1$ and $k_2$ are the circumcircles of the triangles $ACB$ and $ACD$. $k_1$ intersects segments $AD$ and $CD$ at the points $K$ and $L$. $k_2$ intersects segments $AB$ and $DB$ at the points $M$ and $N$. $P$ is the intersection point of $KL$ and $MN$. $X$ is the intersection point of $BP$ and $k_1$. $Y$ is the intersection point of $DP$ and $k_2$. $Z$ and $T$ are the intersection points of $KL$ and $MN$ with $BD$, respectively. Prove that $X$, $Y$, $Z$, $T$ are concyclic.

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I would like to see a beatiful solution of this problem. Do you have any ideas?

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P, C, X, Z are concyclic. P, C, Y, T are concyclic, too. Does it help to solve the problem?

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It is a wrong statement ... it seems to be so close to reality. (At least using software and rounding to the second digit after decimal point shows it is true ... but when we use higher precision it shows it is false statement).