
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 $(xOy)$, the equation of curve $(C_{1})$ is $\frac{x^2}{3} + y^2 = 1$. Establish a polar coordinate system with coordinate origin $O$ as the pole and the positive half of the $x$-axis as the polar axis. The polar equation of curve $(C_{2})$ is $\rho = 4\sin(\theta + \frac{\pi}{3})$. The polar equation of ray $OM$ is $\theta = \alpha_0 (\rho \geqslant 0)$.

(I) Write the polar equation of curve $(C_{1})$ and the cartesian equation of curve $(C_{2})$.

(II) If ray $OM$ bisects curve $(C_{2})$ and intersects curve $(C_{1})$ at point $A$, and point $B$ is another point on curve $(C_{1})$ such that $\angle AOB = \frac{\pi}{2}$, find $|AB|$.

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