
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 \[\begin{cases}x=2+2\cos \varphi \\ y=2\sin \varphi\end{cases} (\varphi \text{ is the parameter})\] and \[\begin{cases}x=\cos \beta \\y=1+\sin \beta\end{cases} (\beta \text{ is the parameter})\] respectively. An polar coordinate system is established with \(O\) as the pole and the positive x-axis as the polar axis.
1. Find the polar coordinate 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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