
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 $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > 0, b > 0)$ with eccentricity $\frac{\sqrt{6}}{3}$, and the area of the quadrilateral formed by its four vertices equals $2\sqrt{3}$.

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

(Ⅱ) A line $l$ passing through the origin with a non-zero slope intersects the ellipse $C$ at points $P$ and $Q$. Let $A$ be the right vertex of the ellipse $C$. The lines $AP$ and $AQ$ intersect the $y$-axis at points $M$ and $N$, respectively. Determine whether the circle with diameter $MN$ always passes through a fixed point on the $x$-axis. If it does, find the coordinates of this fixed point; if not, explain why.

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