(Actually a problem of geometry)

you are given a circle $$$O$$$ whose radius is $$$2$$$.

And there are two points $$$A$$$ and $$$B$$$ out of the circle. it is known that $$$OA = 4, OB = 4\sqrt 2$$$, and also $$$\angle AOB = 45^\circ$$$.

There's another point $$$P$$$ on circle $$$O$$$ (which means that $$$OP = 2$$$).

You are required to write the minimum value of $$$2\sqrt 2PA + PB$$$.

(I guess I may not put the figure here?)