
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
(1) Given $x^{\frac{1}{2}} + x^{-\frac{1}{2}} = 3$, find the value of $x + x^{-1}$.

(2) Calculate the value of $\left( \frac{1}{8} \right)^{-\frac{1}{3}} - 3^{\log_{3}2}(\log_{3}4) \cdot (\log_{8}27) + 2\log_{\frac{1}{6}} \sqrt{3} - \log_{6}2$.

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