Circles $P$, $Q$, and $R$ are externally tangent to one another. The external tangent of circles $P$ and $Q$ that does not intersect circle $R$ intersects circle $P$ at point $P_Q$ and circle $Q$ at point $Q_P$. Points $Q_R$, $R_Q$, $R_P$, and $P_R$ are defined similarly.

If the radius of circle $Q$ is $4$ and the line segment $\overline{Q_PP_Q}$ is parallel to $\overline{R_QQ_R}$, compute the length of $R_PP_R$.

Let's think step by step and output the final answer within \boxed{}.
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