HMMT 2017 G2

Let $A B C$ be a triangle with $A B=13, B C=14$, and $C A=15$. Let $\ell$ be a line passing through two sides of triangle $A B C$. Line $\ell$ cuts triangle $A B C$ into two figures, a triangle and a quadrilateral, that have equal perimeter. What is the maximum possible area of the triangle?