
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 circle $C$ is
$$
\begin{cases}
x = 2\cos\theta \\
y = 2\sin\theta
\end{cases}
$$
where $\theta$ is the parameter. Line $l$ passes through point $P(1,1)$ with a slope angle $\alpha = \frac{\pi}{6}$.
1. Write down the parametric equations of line $l$.
2. Assume line $l$ intersects the circle $C$ at two points $A$ and $B$. Find the product of the distances from point $P$ to points $A$ and $B$.

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