
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
In the Cartesian coordinate system, there is an ellipse C1: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ (where $a>b>0$) with eccentricity $\frac{\sqrt{2}}{2}$. Two perpendicular chords are drawn through the right focus F of the ellipse. When the slope of one chord is $0$, the sum of the lengths of the two chords is $6$.
(1) Find the equation of the ellipse;
(2) Points A and B are on the parabola C2: $x^2 = 4y$, and the tangents at points A and B are perpendicular to each other. The line connecting points A and B intersects the ellipse C1 at points C and D. Find the maximum value of the length of chord CD.

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