
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 that point $P$ is a point on the ellipse $C:\frac{x^{2}}{4}+y^{2}=1$, the tangent line $l$ of the ellipse $C$ at point $P$ intersects the circle $O:x^{2}+y^{2}=4$ at points $A$ and $B$. Find the slope of the tangent line $l$ when the area of triangle $AOB$ is maximized.

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