HMMT November 2019 U13

In $\triangle A B C$, the incircle centered at $I$ touches sides $A B$ and $B C$ at $X$ and $Y$, respectively. Additionally, the area of quadrilateral $B X I Y$ is $\frac{2}{5}$ of the area of $A B C$. Let $p$ be the smallest possible perimeter of a $\triangle A B C$ that meets these conditions and has integer side lengths. Find the smallest possible area of such a triangle with perimeter $p$.