Let $R$ be the $\blue{\text{circular segment}}$ which lies inside the circle $\purple{r=4\cos\theta}$ and to the left of the line $\red{r=\sec\theta}$ as pictured below.
**What is the perimeter of region $R$?**
*Enter the exact answer as an expression involving radicals and $\pi$.*
[[☃ expression 1]]