
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
Taking the origin of the Cartesian coordinate system as the pole and the positive half-axis of the $x$-axis as the polar axis, we establish a polar coordinate system. Both coordinate systems use the same unit of length. It is known that the parametric equation of curve $C_1$ is $\begin{cases} x = \cos \alpha \\ y = \sin \alpha \end{cases}$, where $\alpha$ is the parameter and $\alpha \in [0,\pi)$. The polar equation of curve $C_2$ is $\rho = -2\sin \theta$.

$(1)$ Find the polar equation of $C_1$ and the Cartesian equation of $C_2$;  
$(2)$ If $P$ is any point on $C_1$, and the line $l$ passing through point $P$ intersects $C_2$ at points $M$ and $N$, find the range of values for $|PM| \cdot |PN|$.

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