2019 AIME II Problem 13

Regular octagon A_1A_2A_3A_4A_5A_6A_7A_8 is inscribed in a circle of area 1. Point P lies inside the circle so that the region bounded by \overline{PA_1},\overline{PA_2}, and the minor arc \widehat{A_1A_2} of the circle has area \tfrac{1}{7}, while the region bounded by \overline{PA_3},\overline{PA_4}, and the minor arc \widehat{A_3A_4} of the circle has area \tfrac{1}{9}. There is a positive integer n such that the area of the region bounded by \overline{PA_6},\overline{PA_7}, and the minor arc \widehat{A_6A_7} of the circle is equal to \tfrac{1}{8}-\tfrac{\sqrt2}{n}. Find n.

Leading zeroes must be inputted, so if your answer is 34, then input 034

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.