
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 equation of the curve $C_1$ is given by $\begin{cases} x=1+\cos \alpha \\ y=\sin \alpha \end{cases}$ (where $\alpha$ is the parameter), and the parametric equation of the curve $C_2$ is given by $\begin{cases} x=\cos \beta \\ y=1+\sin \beta \end{cases}$ (where $\beta$ is the parameter). An polar coordinate system is established with the origin $O$ as the pole and the positive $x$-axis as the polar axis.
$(1)$ Find the polar coordinate equations of the curves $C_1$ and $C_2$; 
$(2)$ Given that the ray $l_1: \theta = \alpha\left( \frac{\pi}{6} < \alpha < \frac{\pi}{2}\right)$, rotating ray $l_1$ clockwise by $\frac{\pi}{6}$ results in the ray $l_2: \theta = \alpha - \frac{\pi}{6}$. Also, ray $l_1$ intersects curve $C_1$ at points $O$ and $P$, while ray $l_2$ intersects curve $C_2$ 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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