BALKAN-SL 2012 G4

Let $M$ be the point of intersection of the diagonals of a cyclic quadrilateral $ABCD$ . Let $I_1$ and $I_2$ are the incenters of triangles $AMD$ and $BMC$ , respectively, and let $L$ be the point of intersection of the lines $DI_1$ and $CI_2$ . The foot of the perpendicular from the midpoint $T$ of $I_1I_2$ to $CL$ is $N$ , and $F$ is the midpoint of $TN$ . Let $G$ and $J$ be the points of intersection of the line $LF$ with $I_1N$ and $I_1I_2$ , respectively. Let $O_1$ be the circumcenter of triangle $LI_1J$ , and let $\Gamma_1$ and $\Gamma_2$ be the circles with diameters $O_1L$ and $O_1J$ , respectively. Let $V$ and $S$ be the second points of intersection of $I_1O_1$ with $\Gamma_1$ and $\Gamma_2$ , respectively. If $K$ is point where the circles $\Gamma_1$ and $\Gamma_2$ meet again, prove that $K$ is the circumcenter of the triangle $SVG$ .