
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 an ellipse: $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)$ with one vertex at $A(2, 0)$ and a focal length of $2$. The line $l$ intersects the ellipse at points $E$ and $F$ (both $E$ and $F$ are distinct from $A$) such that $AE \perp AF$.
(I) Find the standard equation of the ellipse;
(II) Let $O$ be the coordinate origin. If point $P$ satisfies $2\overrightarrow{OP} = \overrightarrow{OE} + \overrightarrow{OF}$, find the range of possible slopes for line $AP$.

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