
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 Cartesian coordinate system $xOy$, the parametric equation of the line $l$ is 
$$
\begin{cases}
x=1+t\cos \alpha \\
y=\frac{1}{2}+t\sin \alpha
\end{cases}
(t \text{ is a parameter}),
$$
At the origin $O$ as the pole and the positive $x$-axis as the polar axis, the polar coordinate system is established. The polar equation of the curve $C$ is 
$$\rho^2= \frac{12}{4\sin^2\theta + 3\cos^2\theta}.$$
(1) Write the Cartesian coordinate equation of curve $C$.
(2) Given that point $P$ has Cartesian coordinates $(1, \frac{1}{2})$, line $l$ intersects curve $C$ at two distinct points $A$ and $B$. Determine the range of values for $|PA| \cdot |PB|$.

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