The boundary of the largest area is r = 2, m = 0, and the inner boundary of the right-angle sector is BC, AC, ∠ CoA = 45?

Take any point P (red) in the quadrangle OACB? If there is a route to p, OTP OT = r.tp = m? r+m=2

Extend TP to EF. Note that ⊿FEA∽⊿COA is an isosceles triangle.

2 = OA = OE+EA = OE+EF > OE+EP = OE+ET+TP > OT+TP = r+m = 2，？ Get 2 > 2？ conflict

∴ Quadrilateral OACB is an area that P points cannot reach.

Area of the area reached by point P = π× 2? /4-2×[﹙ 1/2﹚×2×2×sin45？ ]=π-2√2 [? The blue area in the picture? ]