Regular octagon A_{1} A_{2} A_{3} A_{4} A_{5} A_{6} A_{7} A_{8} is inscribed in a circle of area 1. Point P lies inside the circle so that the region bounded by \overline{P A_{1}}, \overline{P A_{2}}, and the minor arc \widehat{A_{1} A_{2}} of the circle has area \frac{1}{7}, while the region bounded by \overline{P A_{3}}, \overline{P A_{4}}, and the minor arc \widehat{A_{3} A_{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{P A_{6}}, \overline{P A_{7}}, and the minor arc \widehat{A_{6} A_{7}} is equal to \frac{1}{8}-\frac{\sqrt{2}}{n}. Find n.