
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 that point \( F \) is the common focus of the parabola \( C: x^{2}=4y \) and the ellipse \( \frac{y^{2}}{a^{2}}+\frac{x^{2}}{b^{2}}=1 \) (\(a > b > 0\)), and the maximum distance from point \( M \) on the ellipse to point \( F \) is 3:

1. Find the equation of the ellipse.
2. Draw two tangent lines from point \( M \) to the parabola \( C \), with points of tangency labeled \( A \) and \( B \). Find the maximum area of \( \triangle MAB \).

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