
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 $a+a^{-1} = \frac{5}{2}$ ($a>1$),  
(1) Find the value of the following expressions:  
   (I) $a^{-\frac{1}{2}} + a^{\frac{1}{2}}$;  
   (II) $a^{\frac{3}{2}} + a^{-\frac{3}{2}}$;  
(2) Given $2\lg(x-2y) = \lg x + \lg y$, find the value of $\log_a \frac{y}{x}$.

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