
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 parametric equations of curve $C_1$ are given by $\begin{cases} x=1+\cos \alpha \\ y=\sin \alpha \end{cases} (\alpha \text{ is the parameter})$, and curve $C_2$ is given by $\frac{x^{2}}{3}+y^{2}=1$.

(1) Find the polar equations of $C_1$ and $C_2$ in the polar coordinate system with $O$ as the pole and the positive $x$-axis as the polar axis.

(2) The ray $\theta = \frac{\pi}{3} (\rho \geqslant 0)$ intersects $C_1$ at point $A$ and $C_2$ at point $B$. Find the distance $|AB|$.

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