
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 $\Gamma$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1(a > b > 0)$ passing through point $A(0,2)$, with an eccentricity of $\frac{\sqrt{2}}{2}$. The line $l$ passing through point $A$ intersects the ellipse at another point $M$.
(I) Find the equation of the ellipse $\Gamma$;
(II) Determine whether there exists a line $l$ such that the circle $C$ with $AM$ as its diameter passes through the right focus $F$ of the ellipse $\Gamma$ and is tangent to the line $x-2y-2=0$. If it exists, find the equation of the line $l$; if not, explain the reason.

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