
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
Simplify and find the value of:<br/>$(1)$ Given $x=\sqrt{3}+\sqrt{5}$ and $y=\sqrt{3}-\sqrt{5}$, find the value of the algebraic expression $2x^{2}-4xy+2y^{2}$.<br/>$(2)$ Simplify first, then find the value of $\left(\frac{{x-y}}{{{x^2}-2xy+{y^2}}}-\frac{x}{{{x^2}-2xy}}\right)÷\frac{y}{{x-2y}}$, where $x=2\sqrt{2}-1$ and $y=2-\sqrt{2}$.

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