
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
Find the equations of the ellipses that satisfy the following conditions:
1. The center of the ellipse is at the origin, the axes are symmetric about the coordinate axes, and it passes through the points $p\_1(\sqrt{6}, 1)$ and $p\_2(-\sqrt{3}, -\sqrt{2})$;
2. The ellipse has axes symmetric about the coordinate axes, the long axis is three times the short axis, and it passes through the point $P(3,0)$.

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