
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 $C$: $\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1(a>b>0)$ with a major axis length of $6$ and eccentricity $e=\frac{1}{3}$. <br/>$(1)$ Find the equation of ellipse $C$; <br/>$(2)$ The left and right foci of ellipse $C$ are $F_{1}$ and $F_{2}$, point $P$ lies on the ellipse, and $\angle PF_{1}F_{2}=60^{\circ}$. Find the area of $\triangle PF_{1}F_{2}$.

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