
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
With the origin of the Cartesian coordinate system as the pole and the positive half-axis as the polar axis, a polar coordinate system is established. The same unit of length is used in both coordinate systems. Suppose point $A$ has polar coordinates $(2, \frac{\pi}{6})$, and line $l$ passes through point $A$ and forms an angle of $\frac{\pi}{3}$ with the polar axis. The polar equation of circle $C$ is $\rho= \sqrt{2}\cos (\theta- \frac{\pi}{4}).$

(Ⅰ) Write the parametric equation of line $l$, and convert the equation of circle $C$ into Cartesian coordinates;  
(Ⅱ) Suppose line $l$ intersects circle $C$ at points $B$ and $C$, calculate the value of $|AB|\cdot|AC|.$

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