
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 plane rectangular coordinate system $xOy$, the parametric equations of curve $C$ are $\left\{\begin{array}{l}x=2+2\cos\alpha\\ y=2+2\sin\alpha\end{array}\right.$ ($\alpha$ is the parameter), and the parametric equations of line $l$ are $\left\{\begin{array}{l}x=t\cos\beta\\ y=t\sin\beta\end{array}\right.$ ($t$ is the parameter, $0\leqslant \beta\ \ \lt \pi$). Establish a polar coordinate system with the origin $O$ as the pole and the positive half-axis of the $x$-axis as the polar axis, using the same unit of length.<br/>$(1)$ Find the polar coordinate equation of curve $C$;<br/>$(2)$ Suppose line $l$ intersects curve $C$ at points $A$ and $B$, and $|OA|^{2}+|OB|^{2}=16$, find the value of $\beta$.

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