
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Three circles with radii \(0<c<b<a\) are tangent to each other and to a line.

(a) Let radii \(a=16\) and \(b=4\). Find the distance \(d\) between the points of tangency of these two circles to the line.

(b) Let radii \(a=16\) and \(b=4\). Find the radius \(c\).

(c) The configuration is called nice if \(a\), \(b\), and \(c\) are all integers. Among all nice configurations, find the smallest possible value of \(c\).

Present the answer in LaTex format: \boxed{Your answer}
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