HMMT 2012 G9

Let $O, O_{1}, O_{2}, O_{3}, O_{4}$ be points such that $O_{1}, O, O_{3}$ and $O_{2}, O, O_{4}$ are collinear in that order, $O O_{1}=$ $1, O O_{2}=2, O O_{3}=\sqrt{2}, O O_{4}=2$, and $\measuredangle O_{1} O O_{2}=45^{\circ}$. Let $\omega_{1}, \omega_{2}, \omega_{3}, \omega_{4}$ be the circles with respective centers $O_{1}, O_{2}, O_{3}, O_{4}$ that go through $O$. Let $A$ be the intersection of $\omega_{1}$ and $\omega_{2}, B$ be the intersection of $\omega_{2}$ and $\omega_{3}, C$ be the intersection of $\omega_{3}$ and $\omega_{4}$, and $D$ be the intersection of $\omega_{4}$ and $\omega_{1}$, with $A, B, C, D$ all distinct from $O$. What is the largest possible area of a convex quadrilateral $P_{1} P_{2} P_{3} P_{4}$ such that $P_{i}$ lies on $O_{i}$ and that $A, B, C, D$ all lie on its perimeter?