
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
Given the Cartesian coordinate system $xOy$, with $O$ as the pole and the non-negative half-axis of $x$ as the polar axis, a polar coordinate system is established. The curve $C$ is described by the parametric equations $\begin{cases} x=2\cos \varphi \\ y=2+2\sin \varphi \end{cases}$ (with $\varphi$ as the parameter). Points $A$ and $B$ are two points on curve $C$, with their polar coordinates being $\left( \rho_1,\frac{\pi}{3} \right)$ and $\left( \rho_2,\frac{5\pi}{6} \right)$ respectively.

$(1)$ Write down the general equation and the polar equation of curve $C$;  
$(2)$ Calculate the value of $|AB|$.

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