Four circles with radii 1, 2, 3, and $r$ are externally tangent to one another, where $r$ is the smallest radius.  Compute $r$.
<|im_start|>system
You are a helpful assistant.<|im_end|>
<|im_start|>user
Four circles with radii 1, 2, 3, and $r$ are externally tangent to one another, where $r$ is the smallest radius.  Compute $r$.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
\frac{6}{23}
Four circles with radii 1, 2, 3, and $r$ are externally tangent to one another, where $r$ is the smallest radius.  Compute $r$.
train