
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 $P(-2,2)$ is on the circle $O$: $x^{2}+y^{2}=r^{2}(r > 0)$, and line $l$ intersects circle $O$ at points $A$ and $B$.  
$(1)r=$ _______ ;  
$(2)$ If $\triangle PAB$ is an isosceles triangle with base $AB=2\sqrt{6}$, find the equation of line $l$.

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