Given a regular nonagon $A_1A_2A_3A_4A_5A_6A_7A_8A_9$ with side length $1$. Diagonals $A_3A_7$ and $A_4A_8$ intersect at point $P$. Find the length of segment $P A_1$.
Source: 2022 Czech-Polish-Slovak Match Junior, team p5 CPSJ
Tags: geometry, nonagon
Given a regular nonagon $A_1A_2A_3A_4A_5A_6A_7A_8A_9$ with side length $1$. Diagonals $A_3A_7$ and $A_4A_8$ intersect at point $P$. Find the length of segment $P A_1$.