
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 $(xOy)$, let $l$ be a line with an angle of inclination $\alpha$ and parametric equations $\begin{cases} x = 3 + t\cos\alpha \\ y = t\sin\alpha \end{cases}$ (where $t$ is a parameter). The line $l$ intersects the curve $C$: $\begin{cases} x = \frac{1}{\cos\theta} \\ y = \tan\theta \end{cases}$ (where $\theta$ is a parameter) at two distinct points $A$ and $B$.

1. If $\alpha = \frac{\pi}{3}$, find the rectangular coordinates of the midpoint of the line segment $AB$.
2. If the slope of line $l$ is 2 and it passes through the known point $P(3, 0)$, find the value of $|PA| \cdot |PB|$.

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