
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<sub>1</sub> is given by
$$
\begin{cases}
x = 3 + 3\cos\theta \\
y = 3\sin\theta
\end{cases},
$$
where θ is the parameter. Taking the origin as the pole and the positive x-axis as the polar axis, the polar equation of curve C<sub>2</sub> is $\rho = \sqrt{3}\sin\theta + \cos\theta$, and the polar equation of curve C<sub>3</sub> is $\theta = \frac{\pi}{3}$.
(Ⅰ) Find the polar equation of curve C<sub>1</sub>.
(Ⅱ) Curve C<sub>3</sub> intersects curve C<sub>1</sub> at points O and A, while curve C<sub>3</sub> intersects curve C<sub>2</sub> at points O and B. Calculate the length of segment AB.

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