
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
The line \( l \) has an inclination angle of \( \frac{\pi}{4} \) and an intercept on the \( y \)-axis of 3. Construct an ellipse with the foci being the foci of the hyperbola \( 12x^2 - 4y^2 = 3 \) such that the ellipse intersects the line \( l \). To minimize the length of the major axis of the ellipse, find the equation of the ellipse.

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