
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
The left and right foci of the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ are $F_1$ and $F_2$, respectively, and its eccentricity is $\frac{1}{2}$. Point $P$ is a moving point on the ellipse, and the maximum area of the inscribed circle of $\triangle F_1PF_2$ is $\frac{\pi}{3}$.

$(1)$ Find the equation of the ellipse;

$(2)$ Suppose the left vertex of the ellipse is $A_1$, and a line $l$ passing through the right focus $F_2$ intersects the ellipse at points $A$ and $B$. Extend $AA_1$ and $A_1B$ to intersect the line $x=4$ at points $P$ and $Q$, respectively. Is the circle with diameter $PQ$ always passing through a fixed point? If yes, find the coordinates of the fixed point; if not, explain why.

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