
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 ellipse that satisfies the following conditions:
1. The coordinates of the two foci are $(-2,0)$ and $(2,0)$, respectively. The sum of the distances from a point $P$ on the ellipse to the two foci equals $6$. Find the equation of the ellipse.
2. The foci of the ellipse are $F_{1}(0,-5)$ and $F_{2}(0,5)$, and the point $P(3,4)$ is on the ellipse. Find the equation of the ellipse.

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