
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
$$
(x-3)^2 + (y-4)^2 = 25
$$
and
$$
(x+6)^2 + (y+5)^2 = 16,
$$
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}
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