2023 AMC 12A Problem 18

Circle C_1 and C_2 each have radius 1, and the distance between their centers is \tfrac12. Circle C_3 is the largest circle internally tangent to both C_1 and C_2. Circle C_4 is internally tangent to both C_1 and C_2 and externally tangent to C_3. What is the radius of C_4?

\textbf{(A)} \dfrac{1}{14} \qquad \textbf{(B)} \dfrac{1}{12} \qquad \textbf{(C)} \dfrac{1}{10} \qquad \textbf{(D)} \dfrac{3}{28} \qquad \textbf{(E)} \dfrac{1}{9}

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