
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 $xOy$, with the origin as the pole and the positive half-axis of $x$ as the polar axis, a polar coordinate system is established. The polar equation of circle $C$ is $\rho=2\sqrt{2}\sin\left(\theta-\frac{\pi}{4}\right)$, and the parametric equation of line $l$ is $\begin{cases} x=-t \\ y=1+t \end{cases}$ with $t$ as the parameter. Line $l$ intersects circle $C$ at points $A$ and $B$.
(Ⅰ) Find the Cartesian coordinate equation of circle $C$;
(Ⅱ) Let $M(0,1)$ be a fixed point on line $l$, find the value of $|MA|\cdot|MB|$.

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