
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 
\[ (x-8)^2 + (y-5)^2 = 49 \]
and 
\[ (x+12)^2 + y^2 = 64 \]
respectively. Find 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