
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]


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Given the parameter equation of line $l$ and the equation of circle $C$ in the polar coordinate system, find the rectangular coordinate equation of circle $C$ and the minimum value of $\frac{1}{|PA|} + \frac{1}{|PB|}$, where $P(1, 2)$ and $A$, $B$ are the intersection points of line $l$ and circle $C$.

The parameter equation of line $l$ in the rectangular coordinate system is $\begin{cases} x = 1 + t\cos\alpha\\ y = 2 + t\sin\alpha \end{cases}$ ($t$ is the parameter), and the equation of circle $C$ in the polar coordinate system (with the same unit length and origin as the rectangular coordinate system, and the positive $x$-axis as the polar axis) is $\rho = 6\sin\theta$.

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