
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 $a + a^{-1} = 4$, find the value of $a^{\frac{1}{2}} + a^{-\frac{1}{2}}$.
2. Given $2\log_{10} \frac{x-y}{2} = \log_{10} x + \log_{10} y$, find the value of $\log_{(3-2\sqrt{2})} \frac{x}{y}$.

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