
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 $C:\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1(a\gt b\gt 0)$ with the right focus $F(c,0)$ and the right vertex $A$, where point $P$ is any point on the ellipse other than point $A$, the maximum area of $\triangle APF$ is $\frac{\sqrt{3}{{b}^{2}}}{6}$.

1. Find the eccentricity of the ellipse $C$.
2. Let the line passing through point $F$ with a slope of $-\frac{3}{4}$ be $l$, and let $Q$ be the intersection point of $l$ and the ellipse above the $x$-axis. Circle $B$ is tangent to both the $x$-axis and line $l$, with center $B$ on the line $x=-4$ and $OB$ parallel to $AQ$. Find the equation of the ellipse $C$.

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