
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 $(xOy)$, the parametric equations of line $l$ are given by: $ \begin{cases} x=1+t\cos \alpha \\ y=2+t\sin \alpha \end{cases} (t$ is the parameter, $0\leqslant \alpha < \pi)$, and the polar coordinate system is established with $O$ as the pole and the positive semi-axis of $x$ as the polar axis. The polar equation of curve $C$ is $\rho=6\sin \theta$.
(I)(i) When $\alpha=\dfrac{\pi}{4}$, write the ordinary equation of line $l$;
(ii) Write the cartesian equation of curve $C$;
(II) If point $P(1, 2)$, and curve $C$ intersects with line $l$ at points $A$ and $B$, find the minimum value of $\dfrac{1}{|PA|}+\dfrac{1}{|PB|}$.

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