
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 equation of line $l$ is $\begin{cases} x=1+t\cos \alpha \\ y=t\sin \alpha \end{cases}$ (where $t$ is the parameter, $0 \leqslant \alpha < \pi$). Taking $O$ as the pole and the positive half-axis of $x$ as the polar axis, a polar coordinate system is established. The polar equation of curve $C$ is $\rho=4\cos \theta$.

(Ⅰ) Find the general equation of line $l$ and the Cartesian equation of curve $C$;  
(Ⅱ) Let $A(1,0)$, line $l$ intersects curve $C$ at points $M$, $N$, and let $P$ be a point on line $l$ such that $\dfrac{2}{|AP|} = \dfrac{1}{|AM|} + \dfrac{1}{|AN|}$. When $|AP|$ is maximized, find the coordinates of point $P$.

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