
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
Given two circles C<sub>1</sub>: $x^{2}+y^{2}=9$ and C<sub>2</sub>: $(x-3)^{2}+(y-4)^{2}=r^{2}$ ($r>0$) are externally tangent to each other.  
(1) If circle C<sub>2</sub> is symmetric about the line $l: \frac{ax}{9}- \frac{by}{12}=1$, find the minimum length of the tangent from point $(a, b)$ to circle C<sub>2</sub>;  
(2) If line $l_{1}$ passes through point A(1, 0) and intersects circle C<sub>2</sub> at points P and Q, find the maximum area of $\triangle C_{2}PQ$ and the equation of line $l_{1}$ at this time.

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