
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, with the coordinate axes as its axes of symmetry, and it passes through two points p<sub>1</sub> ($$\sqrt {6}$$, 1) and p<sub>2</sub> (-$$\sqrt {3}$$, -$$\sqrt {2}$$);  
(2) The ellipse has the coordinate axes as its axes of symmetry, and its major axis is three times the length of its minor axis, and it passes through point P (3, 0).

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