2025 AMC 12A Problem 24

A circle of radius r is surrounded by 12 circles of radius 1, externally tangent to the central circle and sequentially tangent to each other, as shown. Then r can be written as \sqrt{a} + \sqrt{b} + c, where a, b, and c are integers. What is a + b + c?

\textbf{(A)} 3 \qquad \textbf{(B)} 5 \qquad \textbf{(C)} 7 \qquad \textbf{(D)} 9 \qquad \textbf{(E)} 11

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.