
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 an ellipse $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ with eccentricity $e = \frac{\sqrt{2}}{2}$. The point $(1, \frac{\sqrt{2}}{2})$ lies on the ellipse. Let $A$ and $B$ respectively be the right and top vertex of the ellipse. Chords $AE$ and $BF$ of the ellipse $C$ pass through points $A$ and $B$ and intersect the ellipse at points $E$ and $F$.
(1) Determine the equation of the ellipse $C$.
(2) If the slopes of the lines $AE$ and $BF$ are negative reciprocals of each other, find the slope of line $EF$.

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