
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 equations of circles $C_1$ and $C_2$ are respectively $\begin{cases} x=2+2\cos \phi \\ y=2\sin \phi \end{cases}$ (with $\phi$ as the parameter) and $\begin{cases} x=\cos \beta \\ y=1+\sin \beta \end{cases}$ (with $\beta$ as the parameter). Establish a polar coordinate system with $O$ as the pole and the positive $x$-axis as the polar axis, and with the same unit length as the Cartesian system:
$(1)$ Find the polar equations of circles $C_1$ and $C_2$.
$(2)$ The ray $OM: \theta = \alpha (0 < \alpha < \frac{\pi}{2})$ intersects circle $C_1$ at points $O$ and $P$, and intersects circle $C_2$ at points $O$ and $Q$. Find the maximum value of $|OP| \cdot |OQ|$.

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