
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 eccentricity $\frac{\sqrt{6}}{3}$, where $F_1$ and $F_2$ are the left and right foci of the ellipse, respectively, and $A$ is an endpoint of the minor axis of the ellipse with $|AF_1| = \sqrt{6}$.

$(1)$ Find the equation of the ellipse $C$;  
$(2)$ A line $l$ passes through $F_2$ and intersects the ellipse $C$ at points $P$ and $Q$. Find the maximum area of $\triangle PQF_1$.

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