
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
A coordinate system and parametric equations are given in the problem titled "Elective 4-4". In the rectangular coordinate system $xOy$, the parametric equations of the curve $C\_1$ are $x=2+2\cos \alpha$, $y=2\sin \alpha$, where $\alpha$ is a parameter. With the coordinate origin $O$ as the pole and the positive semi-axis of $x$ as the polar axis, establish a polar coordinate system with the same unit length.

(1) Write the polar equation of $C\_1$;

(2) Given the curve $C\_2$: $\dfrac{{{x}^{2}}}{4}+{{y}^{2}}=1$, after stretching and exchanging, curve $C\_3$ is obtained. The ray $\theta =\dfrac{\pi }{3}(\rho > 0)$ intersects with $C\_1$ and $C\_3$ at points $A$ and $B$ respectively. Find $|AB|$.

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