
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 $C_{1}$ and $C_{2}$ are given by $C_{1}$: $(x-2)^{2}+y^{2}=4$ and $C_{2}$: $x^{2}+(y-1)^{2}=1$, respectively. Establish a polar coordinate system with $O$ as the pole and the positive semi-axis of $x$ as the polar axis.
(1) Find the polar equations of circles $C_{1}$ and $C_{2}$;
(2) For the ray $OM$: $\theta=\alpha$, let $O$ and $P$ be the intersection points with circle $C_{1}$ and $O$ and $Q$ be the intersection points with circle $C_{2}$. Find the maximum value of $|OP|\cdot|OQ|$.

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