AMC 10A/12A 2023 (Problem 18/22)

Circle

C_1

and

C_2

each have radius

1

, and the distance between their centers is

\frac12

. 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

\text{(A)}\;\frac{1}{14}

\text{(B)}\;\frac{1}{12}

\text{(C)}\;\frac{1}{10}

\text{(D)}\;\frac{3}{28}

\text{(E)}\;\frac{1}{9}