Balkan 2012 shortlist

Let $M$ be the point of intersection of the diagonals of a cyclic quadrilateral $ABCD$. Let $I_1$ and $I_2$ be the incenters of triangles $AMD$ and $BMC$, respectively, and let $L$ be the point of intersection of the lines $D{I}_1$ and $C{I}_2$. The foot of the perpendicular from the midpoint $T$ of $I_1{I}_2$ 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 $I_{1}N$ and $I_1{I}_2$, respectively. Let $O_1$ be the circumcenter of triangle $L{I}_{1}J$, and let ${\Gamma }_1$ and ${\Gamma }_2$ be the circles with diameters $O_{1}L$ and $O_{1}J$, respectively. Let $V$ and $S$ be the second points of intersection of $I_1{O}_1$ with ${\Gamma }_1$ and ${\Gamma }_2$, respectively. If $K$ is the point where the circles ${\Gamma }_1$ and ${\Gamma }_2$ meet again, prove that $K$ is the circumcenter of the triangle $SVG$.