
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, the parametric equations of curve $C$ are $\left\{\begin{array}{l}x=2+\sqrt{10}\cos\alpha\\ y=\sqrt{10}\sin\alpha\end{array}\right.$ ($\alpha$ is the parameter). 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. The polar coordinate equation of the line $l$ is $\rho \cos \theta +2\rho \sin \theta -12=0$.  
$(1)$ Find the polar coordinate equation of curve $C$;  
$(2)$ Let the ray $l_{1}: \theta=\frac{\pi}{4}(\rho\geq0)$ intersect curve $C$ at point $A$ and line $l$ at point $B$. Find the value of $|AB|$.

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