
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 parametric equation of curve $C$ is $\begin{cases}x=2\cos \phi \\ y=2+2\sin \phi\end{cases}(\phi \text{ is the parameter}).$ Points $A$ and $B$ are two points on curve $C$, and the polar coordinates of points $A$ and $B$ are respectively $(\rho_{1}, \frac{\pi}{3}),(\rho_{2}, \frac{5\pi}{6}).$

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