
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 ellipses that satisfy the following conditions respectively.

(1) Passes through the point $(3,-2)$ and has the same foci as the ellipse $4x^2 + 9y^2 = 36$.

(2) The center is at the origin, the foci are on the $x$-axis, the eccentricity is $\frac{\sqrt{2}}{2}$, and the perimeter of the triangle formed by the right focus and the intersection points of the ellipse and the line passing through the left focus is $16$.

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