Largest Isosceles Right Triangle Around a Fixed Right Triangle

A right triangle $PQR$ with legs $1$ and $2$ (hypotenuse $\sqrt5$) is inscribed in an isosceles right triangle $ABC$ whose right angle is at $C$, so that $AC=BC$ and $AB$ is the hypotenuse. The right-angle vertex $P$ of $\triangle PQR$ lies on $AB$, while $Q$ lies on $BC$ and $R$ lies on $CA$.

Find the maximum possible area of $\triangle ABC$.