
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 plane Cartesian coordinate system $xOy$, the parametric equations of the line $l$ are $\left\{\begin{array}{l}x=1-\frac{\sqrt{2}}{2}t\\ y=2+\frac{\sqrt{2}}{2}t\end{array}\right.$ (where $t$ is the parameter). Taking the coordinate origin $O$ as the pole and the positive half of the $x$-axis as the polar axis, establish a polar coordinate system. The polar coordinate equation of the curve $C$ is $\rho =4\sin \theta$.  
$(1)$ Find the general equation of the line $l$ and the rectangular coordinate equation of the curve $C$;  
$(2)$ Given the point $M(1,2)$, the line $l$ and the curve $C$ intersect at points $A$ and $B$, find the value of $|MA|\cdot |MB|$.

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