
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 $xOy$, the parametric equations of curve $C_{1}$ are $\left\{{\begin{array}{l}{x=1+t}\\{y=1+\sqrt{2}t}\end{array}}\right.$ (where $t$ is a parameter). Establish a polar coordinate system with the origin as the pole and the positive $x$-axis as the polar axis. The polar coordinate equation of curve $C_{2}$ is $\rho \left(1-\sin \theta \right)=1$.  
$(1)$ Find the general equation of curve $C_{1}$ and the Cartesian equation of curve $C_{2}$.  
$(2)$ Let $M(1,1)$, the intersection points of curves $C_{1}$ and $C_{2}$ be $A$ and $B$. Find the value of $|MA|\cdot |MB|$.

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