
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
Given in the Cartesian coordinate system $xOy$, with $O$ as the pole and the positive half-axis of $x$ as the polar axis, the parametric equation of curve $C_{1}$ is: 
$$
\begin{cases} 
x=1+\frac{\sqrt{2}}{2}t \\ 
y=-2+\frac{\sqrt{2}}{2}t \\ 
\end{cases},
$$
and the polar equation of curve $C_{2}$ is: $\rho^{2}(1+\sin^{2}\theta)=8$,

$(1)$ Write the standard equations of $C_{1}$ and $C_{2}$;

$(2)$ If $C_{1}$ intersects $C_{2}$ at two points $A$, $B$, find the value of $|AB|$.

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