
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 circle $C$ are given by $\begin{cases} x=3+2\cos \alpha \\ y=2\sin \alpha \end{cases} (\alpha \text{ is the parameter})$. Establish a polar coordinate system $(r, \theta)$ with the origin as the pole and the positive $x$-axis as the polar axis. The polar equation of line $l$ is given by $\rho\cos \theta+\rho\sin \theta+1=0$.

1. Write the standard form equation for circle $C$.
2. Convert the polar equation of line $l$ to rectangular coordinates.
3. Through any point $P$ on line $l$, draw a line intersecting circle $C$ at points $A$ and $B$. Find the minimum value of $|PA| \cdot |PB|$.

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