Two circles $A$ and $B$, both with radius $1$, touch each other externally. Four circles $P$, $Q$, $R$, and $S$, all with the same radius $r$, are such that $P$ touches $A$, $B$, $Q$, $S$ externally; $Q$ touches $P$, $B$, and $R$ externally; $R$ touches $A$, $B$, $Q$, and $S$ externally; and $S$ touches $P$, $A$, and $R$ externally. Calculate $r$.
Context: The question above is from the 2011 IMC (International Mathematics Competition) exam. The solutions do not exist anywhere on the internet. Personally, I believe that the solution to this question will help the geometrical thought of the users of this website.
Attempt: I tried solving the question above by connecting the radii; however, my efforts did not prosper.
Can you guys please help me?