
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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