
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
Let \(C_1\) and \(C_2\) be circles defined by 
\[
(x-12)^2 + y^2 = 25
\]
and 
\[
(x+18)^2 + y^2 = 100,
\]
respectively. What is the length of the shortest line segment \(\overline{PQ}\) that is tangent to \(C_1\) at \(P\) and to \(C_2\) at \(Q\)?

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