
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 a line $l$ passes through the fixed point $P(1,1)$ and has an inclination angle of $\frac{\pi}{4}$. In the coordinate system with the origin as the pole and the positive half-axis of $x$ as the polar axis, the polar equation of curve $C$ is $\rho = 2\cos \theta + \frac{3}{\rho}$.

$(1)$ Find the Cartesian equation of curve $C$ and the parametric equation of line $l$;

$(2)$ If line $l$ intersects curve $C$ at two distinct points $A$ and $B$, find the value of $|AB|$ and $|PA| \cdot |PB|$.

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