
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 $\dfrac {x^{2}}{a^{2}}+ \dfrac {y^{2}}{b^{2}}=1(a > b > 0)$ are $F_{1}$ and $F_{2}$, respectively, and its eccentricity is $\dfrac {1}{2}$. Point $P$ is a moving point on the ellipse, and the maximum area of the inscribed circle of $\triangle F_{1}PF_{2}$ is $\dfrac {\pi}{3}$.  
$(1)$ Find the equation of the ellipse;  
$(2)$ Suppose the left vertex of the ellipse is $A_{1}$, a line $l$ passing through the right focus $F_{2}$ intersects the ellipse at points $A$ and $B$. Extend $A_{1}A$ and $A_{1}B$ to intersect the line $x=4$ at points $P$ and $Q$, respectively. Is the circle with $PQ$ as its diameter 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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