
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 $xOy$, let the parametric equation for curve $C_1$ be $\begin{cases} x=\cos \theta \\ y=1+\sin \theta \end{cases}$ (where $\theta$ is the parameter), and let the equation for curve $C_2$ be $x^2+ \frac{y^2}{2}=1$. Establish a polar coordinate system with the origin $O$ as the pole and the non-negative $x$-axis as the polar axis, maintaining the same unit lengths as in the Cartesian coordinate system $xOy$.

$(1)$ Find the polar equations of curve $C_1$ and curve $C_2$.

$(2)$ Let the ray $\theta= \frac{\pi}{3}$ (where $\rho > 0$) intersect curve $C_1$ at point $A$, distinct from the pole, and intersect curve $C_2$ at point $B$. Calculate $|AB|$.

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