
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 that $a+a^{-1}=\frac{5}{2} (a>1)$, find the values of the following expressions:
(1) $a^{-\frac{1}{2}}+a^{\frac{1}{2}}$;
(2) $a^{\frac{3}{2}}+a^{-\frac{3}{2}}$;
(2) If $2\log(x-2y)=\log x + \log y$, find the value of $\log_{a} \frac{y}{x}$.

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