
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 rectangular coordinate system $xOy$, the line $l$ passes through point $P(1,0)$ with an angle of inclination of $\frac{\pi}{6}$. Establish a polar coordinate system with the coordinate origin $O$ as the pole and the positive semi-axis of the $x$-axis as the polar axis. The polar equation of the curve $C$ is $\rho = 4\cos(\theta + \frac{\pi}{3})$.

1. Write the parametric equation of line $l$ and the rectangular equation of curve $C$.
2. Suppose line $l$ intersects curve $C$ at points $A$ and $B$. Find the value of $|PA| + |PB|$.

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