
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 rectangular coordinate system $xOy$, given the parametric equations of the line $l$ as $\left\{\begin{array}{l}{x=-1-t}\\{y=2+t}\end{array}\right.$ (where $t$ is a parameter), with the origin as the pole and the positive $x$-axis as the polar axis, the polar coordinate equation of curve $C$ is ${\rho^2}=\frac{2}{{1+{{\sin}^2}\theta}}$. Line $l$ intersects curve $C$ at points $A$ and $B$.  
$(1)$ Find the general equation of line $l$ and the rectangular coordinate equation of curve $C$.  
$(2)$ Given that the polar coordinates of point $P$ are $({\frac{{\sqrt{2}}}{2},\frac{π}{4}})$, find the value of $|PA|\cdot |PB|$.

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