
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 $$\begin{cases} \overset{x=2+2\cos\phi }{y=2\sin\phi }\end{cases}$$ (where $\phi$ is the parameter) and $$\begin{cases} \overset{x=\cos\beta }{y=1+\sin\beta }\end{cases}$$ (where $\beta$ is the parameter). Establish a polar coordinate system with $O$ as the pole and the positive half-axis of $x$ as the polar axis.  
(1) Find the polar equations of circles $C_1$ and $C_2$;  
(2) The ray $OM: \theta=\alpha$ 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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