
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 curve equation that satisfies the following conditions:<br/>$(1)$ Given that the focus of the hyperbola is on the $x$-axis, the eccentricity is $\frac{5}{3}$, and the distance between the two vertices is $6$, find the standard equation of the hyperbola;<br/>$(2)$ Given that the length of the major axis of the ellipse is three times the length of the minor axis, passing through point $A(3,0)$, and symmetric with the coordinate axes, find the standard equation of the ellipse.

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