
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 a circle $C: (x-2)^{2}+y^{2}=9$.  
$(1)$ Line $l_{1}$ passes through point $D(-1,1)$ and is tangent to circle $C$. Find the equation of line $l_{1}$.  
$(2)$ Line $l_{2}: x+\sqrt{3}y-1=0$ intersects circle $C$ at points $M$ and $N$. Point $P$ is a moving point on circle $C$. Find the maximum area $S$ of triangle $PMN$.

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