
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 line $l$ is $\begin{cases}x=-1+t\cos \alpha \\ y= \frac{1}{2}+t\sin \alpha\end{cases}$ (where $t$ is the parameter), with the origin $O$ as the pole and the positive half-axis of $x$ as the polar axis, a polar coordinate system is established. The polar equation of curve $C$ is ${\rho}^{2}= \frac{4}{4{\sin }^{2}\theta+{\cos }^{2}\theta}$.

$(1)$ Write the Cartesian coordinate equation of curve $C$;

$(2)$ Given that the Cartesian coordinates of point $P$ are $\left(-1, \frac{1}{2}\right)$, line $l$ intersects curve $C$ at two distinct points $A$ and $B$. Find the range of values for $|PA|\cdot|PB|$.

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