
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
A square is inscribed in the ellipse whose equation is $x^2 + 3y^2 = 3$. One vertex of the square is at $(0, 1)$, and one diagonal of the square lies along the y-axis. Determine the square of the length of each side of the square.

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