
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 the ellipse \(C_1: \frac{x^2}{4} + \frac{y^2}{3} = 1\) and the parabola \(C_2: y^2 = 4x\). From a point \(P\) (not the origin \(O\)) on the parabola \(C_2\), a tangent line \(l\) is drawn. The line \(l\) intersects the ellipse \(C_1\) at points \(A\) and \(B\).

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

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