
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 $C$ with its center at the origin, foci on the x-axis, eccentricity $\frac{\sqrt{3}}{2}$, and the area of the triangle formed by any three vertices of the ellipse being $\frac{1}{2}$:
(1) Find the equation of the ellipse $C$;
(2) If a line $l$ passing through the point $P(\lambda, 0)$ intersects the ellipse at 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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