
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 a Cartesian coordinate system, a line $l$ passes through point $P(2, \sqrt {3})$ and has an angle of inclination $\alpha$. Using the origin as the pole and the positive semi-axis of the x-axis as the polar axis, a polar coordinate system is established. The polar equation of curve $C$ is $\rho = 4 \cos(\theta - \frac{\pi}{3})$. The line $l$ intersects curve $C$ at points $A$ and $B$.  
(1) Find the Cartesian coordinate equation of curve $C$.  
(2) If $|AB| = \sqrt {13}$, find the value of the angle of inclination $\alpha$ of line $l$.

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