
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 parameteric equation of curve $({C}_{1})$ is given by $\begin{cases}& x=1+\cos \alpha \;& y=\sin \alpha \end{cases}$ ($\alpha$ is the parameter), and the curve $({C}_{2}: \frac{{x}^{2}}{3}+{y}^{2}=1)$.
(I) In the polar coordinate system with $O$ as the pole and the positive half of the $x$-axis as the polar axis, find the polar equations of $({C}_{1},{C}_{2})$;
(II) The ray $\theta = \frac{\pi }{3}(\rho \geqslant 0)$ intersects $({C}_{1})$ at point $A$ and $({C}_{2})$ at point $B$, find $|AB|$.

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