
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 standard equations of the following curves:

1. An ellipse with a major axis length of 12, an eccentricity of $\frac{2}{3}$, and a focus on the $x$-axis.
2. The standard equation of the ellipse passing through points $A(\frac{\sqrt{6}}{3}, \sqrt{3})$ and $B(\frac{2\sqrt{2}}{3}, 1)$.

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