Regions in an Octagon

A regular octagon $A_1A_2A_3A_4A_5A_6A_7A_8$ is inscribed in a circle of area $1$. A point $P$ inside the circle is such that the region bounded by $PA_1$, $PA_2$, and the minor arc $A_1A_2$ has area $\dfrac 17$, and the region bounded by $PA_3$, $PA_4$, and the minor arc $A_3A_4$ has area $\dfrac 19$. The region bounded by $PA_6$, $PA_7$, and the minor arc $A_6A_7$ has area $\dfrac 18-\dfrac{\sqrt 2}n$ for a positive integer $n$. Find $n$.