
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 rectangular coordinate system $(xOy)$, the parametric equations of circles $C_{1}$ and $C_{2}$ are given by $\begin{cases} x=2+2\cos \phi \\ y=2\sin \phi \end{cases} (\phi \text{ is the parameter})$ and $\begin{cases} x=\cos \phi \\ y=1+\sin \phi \end{cases} (\phi \text{ is the parameter})$ respectively. Establish a polar coordinate system with $O$ as the pole and the positive half of the $x$-axis as the polar axis.

1. Find the polar equations of circles $C_{1}$ and $C_{2}$.
2. The ray $OM: \theta = a$ 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}
user