
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, line $l$ passes through point $P(1,1)$ with an inclination angle $α= \frac{\pi}{6}$. Now, with the origin of the Cartesian coordinate system as the pole and the non-negative half-axis of the $x$-axis as the polar axis, establish a polar coordinate system. Suppose the polar equation of curve $C$ is $\rho\sin^{2}\theta = 8\cos\theta$.  
(1) Write the parametric equation of line $l$ and the Cartesian coordinate equation of curve $C$;  
(2) Suppose line $l$ intersects curve $C$ at points $A$ and $B$. Calculate the value of $|PA| \cdot |PB|$.

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