
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
Let ellipse $C$: $\dfrac {x^{2}}{a^{2}}+ \dfrac {y^{2}}{b^{2}}=1$ have its right focus at $F_{1}$, its left focus at $F_{2}$, and its top vertex at $A$. A line passing through $A$ and perpendicular to $AF_{2}$ intersects the negative half of the $x$-axis at point $Q$, and $2 \overrightarrow{F_{1}F_{2}}+ \overrightarrow{F_{2}Q}=0$.

$(1)$ Find the eccentricity of ellipse $C$;

$(2)$ If the circle passing through points $A$, $Q$, and $F_{2}$ is tangent to the line $x- \sqrt {3}y-3=0$, find the equation of ellipse $C$;

$(3)$ Under the condition of $(2)$, a line passing through the right focus $F_{2}$ intersects the ellipse at points $M$ and $N$. Given point $P(4,0)$, find the maximum area of $\triangle PMN$.

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