
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
In the rectangular coordinate system $xOy$, an ellipse $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ has an eccentricity of $\frac{\sqrt{2}}{2}$, and the distance between the right focus $F$ and the left directrix is $6\sqrt{2}$.

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

(2) Let $A$ be the left vertex of ellipse $C$, $P$ be a point on ellipse $C$ above the $x$-axis, line $PA$ intersects the $y$-axis at point $M$, and the line perpendicular to $MF$ through point $F$ intersects the $y$-axis at point $N$.

   (i) When the slope of line $PA$ is $\frac{1}{2}$, find the equation of the circumcircle of $\triangle MFN$;

   (ii) Let line $AN$ intersect ellipse $C$ at another point $Q$, find the maximum area of $\triangle PAQ$.

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