
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 $x=\frac{\sqrt{11}}{2}+\frac{\sqrt{7}}{2}$ and $y=\frac{\sqrt{11}}{2}-\frac{\sqrt{7}}{2}$, find the values of $x+y$ and $xy$. Then, using the results from part $(1)$, find the values of the following algebraic expressions: $①x^{2}y+xy^{2}$ and $②\frac{y}{x}+\frac{x}{y}$.

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