
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
Given the ellipse \(b^{2} x^{2} + a^{2} y^{2} = a^{2} b^{2}\), draw a rectangle inside it with a given area \(t\). Determine the coordinates of the vertices of this rectangle. Also, determine the coordinates of the vertices of the rectangle with the largest possible area that can be inscribed in the ellipse.

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