
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
In the polar coordinate system, let curve \(C_1\) be defined by \(\rho\sin^2\theta = 4\cos\theta\). Establish a Cartesian coordinate system \(xOy\) with the pole as the origin and the polar axis as the positive \(x\)-axis. The curve \(C_2\) is described by the parametric equations: 
\[ \begin{cases} 
x = 2 + \frac{1}{2}t \\ 
y = \frac{\sqrt{3}}{2}t 
\end{cases} \]
where \(t\) is the parameter.
(1) Find the Cartesian equations for \(C_1\) and \(C_2\).
(2) If \(C_1\) and \(C_2\) intersect at points \(A\) and \(B\), and there is a fixed point \(P\) with coordinates \((2,0)\), find the value of \(|PA| \cdot |PB|\).

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