
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 $\Omega: \dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1 (a > b > 0)$ with a focal length of $2\sqrt{6}$ and a short axis length of $2\sqrt{2}$.
1. Find the equation of $\Omega$;
2. If the line $y=x+2$ intersects $\Omega$ at points $A$ and $B$, find the standard equation of the circle with diameter $AB$.

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