
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
Determine the standard equations for the following curves:
(1) An ellipse with a major axis length of 12, an eccentricity of $\frac{2}{3}$, and foci on the x-axis;
(2) The standard equation of an ellipse passing through points A $\left(\frac{\sqrt{6}}{3}, \sqrt{3}\right)$ and B $\left(\frac{2\sqrt{2}}{3}, 1\right)$.

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