
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 $C$: $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\left(a \gt b \gt 0\right)$ with the right vertex $A(2,0)$, $P$ is a moving point on the ellipse $C$ and not on the $x$-axis, $O$ is the origin, and the maximum area of $\triangle AOP$ is $1$.  
$(Ⅰ)$ Find the equation of the ellipse $C$ and its eccentricity.  
$(Ⅱ)$ The line passing through point $H(-1,0)$ intersects the ellipse $C$ at another point $Q$, lines $AP$ and $AQ$ intersect the $y$-axis at points $E$ and $F$ respectively. When $|EF|=2$, find the equation of the line $PH$.

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