
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 \(\begin{cases} x=1+\cos \alpha \\ y=\sin \alpha \end{cases}\) (\(\alpha\) is the parameter), and the curve \(C_2: \dfrac{x^2}{3}+y^2=1\).

(1) In the polar coordinate system with the origin \(O\) as the pole and the positive x-axis as the polar axis, find the polar equations for \(C_1\) and \(C_2\).

(2) The ray \(\theta =\dfrac{\pi}{3} (\rho \geqslant 0)\) intersects with \(C_1\) at a point \(A\) other than the pole, and intersects with \(C_2\) at point \(B\), find \(|AB|\).

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