Question 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 $\overset{\frown}{A_1A_2}$ of the circle has area $\frac{1}{7},$ while the region bounded by $\overline{PA_3},$ $\overline{PA_4},$ and the minor arc $\overset{\frown}{A_3A_4}$ of the circle has area $\frac{1}{9}.$ 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 $\overset{\frown}{A_6A_7}$ of the circle is equal to $\frac{1}{8} - \frac{\sqrt{2}}{n}.$ Find $n.$