
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 $xOy$, point $P\left( \frac{1}{2},1\right)$ is given. The parametric equation of line $l$ is $\begin{cases}x= \frac{1}{2}+ \frac{\sqrt{3}}{2}t \\ y=1+ \frac{1}{2}t\end{cases}$ (where $t$ is the parameter). If $O$ is taken as the pole and $Ox$ as the polar axis, and the same unit length is chosen to establish the polar coordinate system, then the polar equation of curve $C$ is $\rho= \sqrt{2}\cos \left(\theta- \frac{\pi}{4}\right)$  
(Ⅰ) Find the general equation of line $l$ and the Cartesian equation of curve $C$;  
(Ⅱ) Suppose line $l$ intersects curve $C$ at points $A$ and $B$, find the product of the distances from point $P$ to points $A$ and $B$.

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