
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 $(-1,\frac{1}{2})$, and line $l$ intersects curve $C$ at two different 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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