
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
An equilateral triangle is inscribed in the ellipse given by $x^2 + 3y^2 = 3$. One vertex of the triangle is at $(0, \sqrt{3}/3)$, and one altitude of the triangle is along the y-axis. Determine the square of the length of each side of the triangle in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, and find $m+n$.

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