HMMT November 2024 U8

Derek is bored in math class and is drawing a flower. He first draws 8 points $A_{1}, A_{2}, \ldots, A_{8}$ equally spaced around an enormous circle. He then draws 8 arcs outside the circle where the $i$ th arc for $i=1,2, \ldots, 8$ has endpoints $A_{i}, A_{i+1}$ with $A_{9}=A_{1}$, such that all of the arcs have radius 1 and any two consecutive arcs are tangent. Compute the perimeter of Derek's 8-petaled flower.