
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 point $A(2,0)$, line $4x+3y+1=0$ is intercepted by circle $C$: $(x+3)^{2}+(y-m)^{2}=13(m < 3)$ to form a chord of length $4\sqrt{3}$, and $P$ is any point on circle $C$.
(1) Find the maximum and minimum values of $|PA|$.
(2) Circle $C$ intersects the coordinate axis at three points. Find the radius of the inscribed circle of the triangle formed by these three points.

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