
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 rectangular coordinate system $xOy$, the parametric equations of line $l$ are given by $\begin{cases} x=t\cos\alpha \\ y=2+t\sin\alpha \end{cases}$ ($t$ is the parameter). Establish a polar coordinate system with the coordinate origin as the pole and the positive semi-axis of $x$ as the polar axis. The polar equation of curve $C$ is $\rho^2 + 2\rho\cos\theta - 2\rho\sin\theta + 1 = 0$.

(I) Find the Cartesian equation of line $l$ and the Cartesian equation of curve $C$ when $\alpha = \frac{\pi}{4}$.

(II) If line $l$ intersects curve $C$ at points $A$ and $B$, and the angle of inclination of line $l$ is $\alpha \in (0, \frac{\pi}{3}]$, point $P$ is the intersection of line $l$ and the $y$-axis. Find the minimum value of $\frac{|PA|\cdot |PB|}{|PA| + |PB|}$.

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