HMMT 2006 F3

Let $C$ be the unit circle. Four distinct, smaller congruent circles $C_{1}, C_{2}, C_{3}, C_{4}$ are internally tangent to $C$ such that $C_{i}$ is externally tangent to $C_{i-1}$ and $C_{i+1}$ for $i=$ $1, \ldots, 4$ where $C_{5}$ denotes $C_{1}$ and $C_{0}$ represents $C_{4}$. Compute the radius of $C_{1}$.