
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 Cartesian coordinate system, let the curve $C_1$ be defined by the equation $x^{2}-y^{2}=2$, and let the curve $C_2$ have the parametric equations 
$$ \begin{cases} x=2+2\cos \theta, \\ y=2\sin \theta \end{cases} (\theta \text{ is the parameter}). $$ 
Construct a polar coordinate system with the origin $O$ as the pole and the positive half of the $x$-axis as the polar axis.

(I) Derive the polar equations of curves $C_1$ and $C_2$.
(II) In the polar coordinate system, the ray $\theta= \frac{\pi}{6}$ intersects curves $C_1$ and $C_2$ at points $A$ and $B$ (different from the pole $O$), respectively. Given the fixed point $M(3,0)$, find the area of triangle $\triangle MAB$.

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