
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 the center of the ellipse \( C \) is at the origin, the foci are on the \( x \)-axis, and the eccentricity is \(\frac{\sqrt{3}}{2}\). The area of the triangle formed by any three vertices of the ellipse is \(\frac{1}{2}\).

(1) Find the equation of the ellipse \( C \).  
(2) If a line \( l \) passes through the point \( P(\lambda, 0) \) and intersects the ellipse at two distinct points \( A \) and \( B \), and \(\overrightarrow{A P} = 2 \overrightarrow{P B}\), determine the range of the real number \(\lambda\).

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