
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 equation of the curve that satisfies the following conditions:
1. The major axis of the ellipse is on the $x$-axis, the length of the major axis is $12$, and the eccentricity is $\frac{2}{3}$;
2. Find the standard equation of the hyperbola that shares the same foci with the hyperbola $\frac{x^2}{16}-\frac{y^2}{9}=1$ and passes through the point $P\left(-\frac{\sqrt{5}}{2},-\sqrt{6}\right)$.

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