HMMT 2020 G6

Let $A B C$ be a triangle with $A B=5, B C=6, C A=7$. Let $D$ be a point on ray $A B$ beyond $B$ such that $B D=7, E$ be a point on ray $B C$ beyond $C$ such that $C E=5$, and $F$ be a point on ray $C A$ beyond $A$ such that $A F=6$. Compute the area of the circumcircle of $D E F$.