
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
Ellipse \( C_{1}: \frac{x^{2}}{4}+\frac{y^{2}}{3}=1 \) and parabola \( C_{2}: y^{2}=4x \) are given. A tangent \( l \) is drawn from a point \( P \) on parabola \( C_{2} \) (other than the origin \( O \)), intersecting the ellipse \( C_{1} \) at two points \( A \) and \( B \).

Find:
(1) The range of the \( x \)-intercept of the tangent \( l \).
(2) The maximum area \( S \) of triangle \( AOB \).

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