
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
(For this question, full marks are 10 points) Compulsory 4-4: Polar Coordinates and Equations  
In the Cartesian coordinate system xOy, the curve C<sub>1</sub> is given by a parametric equation (with parameter $\alpha$). Point M is a moving point on C<sub>1</sub>, and point P satisfies OP = 2OM. The trajectory of point P is the curve C<sub>2</sub>.  
(I) Find the equation of C<sub>2</sub>;  
(II) In the polar coordinate system with O as the pole and the positive x-axis as the polar axis, the ray $\theta =$ intersects with C<sub>1</sub> at a point A (different from the pole), and intersects with C<sub>2</sub> at a point B (different from the pole). Find $|AB|$.

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