
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 the ellipse $\Omega$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ with a focal length of $2\sqrt{3}$, and it passes through the point $(1, \frac{\sqrt{3}}{2})$.
(I) Find the equation of the ellipse $\Omega$;
(II) Let $A$ be the intersection point of the ellipse $\Omega$ and the positive semi-axis of $y$. Are there two points $M$ and $N$ on the ellipse $\Omega$ such that $\triangle AMN$ is an isosceles right triangle with $A$ as the right angle vertex? If so, please explain how many there are; if not, please explain the reason.

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