
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 $\dfrac{x^{2}}{a^{2}} + \dfrac{y^{2}}{b^{2}} = 1 (a > b > 0)$ with its left focus at $F(-c,0)$, the right vertex at $A$, and the coordinates of point $E$ as $(0,c)$, the area of $\triangle EFA$ is $\dfrac{b^{2}}{2}$.  
$(I)$ Find the eccentricity of the ellipse;  
$(II)$ Suppose point $Q$ is on segment $AE$, $|FQ| = \dfrac{3}{2}c$. Extend segment $FQ$ to intersect the ellipse at points $P$, and let points $M$, $N$ be on the $x$-axis, with $PM \parallel QN$, and the distance between lines $PM$ and $QN$ is $c$, the area of quadrilateral $PQNM$ is $3c$.  
$(i)$ Find the slope of line $FP$;  
$(ii)$ Find the equation of the ellipse.

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