
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 $(xOy)$, establish a polar coordinate system with $O$ as the pole and the positive semi-axis of $x$ as the polar axis. The parametric equation of the curve $C$ is given by $\begin{cases} x = 2 \cos \phi, \\ y = 2 + 2 \sin \phi \end{cases}$ ($\phi$ is the parameter). Points $A$ and $B$ are two points on the curve $C$, and their polar coordinates are $\left(\rho, \frac{\pi}{3} \right)$ and $\left(\rho_2, \frac{5\pi}{6} \right)$ respectively. Find the distance between points $A$ and $B$, denoted by $|AB|$.

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