
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
Calculate:<br/>$(1)$ Given ${a}^{\frac{1}{2}}+{a}^{-\frac{1}{2}}=3$, find the value of $\frac{{a}^{\frac{3}{2}}+{a}^{-\frac{3}{2}}+2}{{a}^{2}+{a}^{-2}+3}$.<br/>$(2)$ Given $\lg x+\lg y=2\lg \left(y-2x\right)$, find the value of $log_{\sqrt{2}}\frac{x}{y}$.

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