
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 equation of curve $C_1$ is $\begin{cases} x=\cos \theta \\ y=1+\sin \theta \end{cases}$ (where $\theta$ is the parameter), and the equation of curve $C_2$ is $\frac{x^2}{1}+\frac{y^2}{2}=1$. With $O$ as the pole and the non-negative half-axis of $x$ as the polar axis, a polar coordinate system is established with the same unit of length as the Cartesian coordinate system $xOy$.

(1) Find the polar equations of curves $C_1$ and $C_2$;

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

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