
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
Given in the Cartesian coordinate system $xOy$, the parametric equations of curve $C$ are
$$
\begin{cases}
x = \cos \theta, \\
y = \sin \theta
\end{cases}
$$
with $\theta$ as the parameter. A line $l$ passes through the fixed point $P(1,1)$ with an angle of inclination $\frac{\pi}{6}$.

(I) Write the parametric equations for line $l$. If the x-coordinates of the conic curve $C$ are elongated to twice their original length and the y-coordinates remain unchanged, find the standard equation for this new curve $C'$.

(II) Suppose line $l$ intersects the conic curve $C$ at points $A$ and $B$. Compute the product of the lengths $|PA| \cdot |PB|$.

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