
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 the value of the following expressions:  
(1) If $a+a^{-1}=4$, then find the value of $a^{\frac{1}{2}}+a^{-\frac{1}{2}}$.  
(2) Given $2\lg\left(\frac{x-y}{2}\right)=\lg x+\lg y$, find the value of $\log_{(3-2\sqrt{2})}\left(\frac{x}{y}\right)$.

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