
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 equations of circles C<sub>1</sub> and C<sub>2</sub> are C<sub>1</sub>: $(x-2)^2+y^2=4$ and C<sub>2</sub>: $x^2+(y-1)^2=1$. 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<sub>1</sub> and C<sub>2</sub>;  
(2) The ray OM: $\theta=\alpha$ intersects circle C<sub>1</sub> at points O and P, and intersects circle C<sub>2</sub> 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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