
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 $\begin{cases}x=-1+t\cos \alpha \\ y=t\sin \alpha\end{cases}$ (where $t$ is the parameter, and $\alpha$ is the inclination angle of the line). Taking the origin of the Cartesian coordinate system $xOy$ as the pole and the positive half-axis of $x$ as the polar axis, the same unit of length is used to establish the polar coordinate system. The polar equation of circle $C$ is $\rho=2\cos \theta$. Suppose line $l$ intersects circle $C$ at points $A$ and $B$.

$(1)$ Find the Cartesian coordinate equation of circle $C$ and the range of values for $\alpha$;

$(2)$ If the coordinates of point $P$ are $(-1,0)$, find the range of values for $\dfrac{1}{|PA|}+\dfrac{1}{|PB|}$.

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