
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 $x=\frac{1}{3+2\sqrt{2}}$ and $y=\frac{1}{3-2\sqrt{2}}$.

$(1)$ Find the value of $x^{2}+y^{2}+xy$.

$(2)$ If the decimal part of $x$ is $m$ and the decimal part of $y$ is $n$, find the value of $(m+n)^{2023}-\sqrt[3]{(m-n)^3}$.

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