
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 following expressions:
1. $(1)(2a^{\frac{2}{3}}b^{\frac{1}{2}})(-6a^{\frac{1}{2}}b^{\frac{1}{3}}) \div (-3a^{\frac{1}{6}}b^{\frac{5}{6}}) (a > 0, b > 0)$
2. $2(\lg{\sqrt{2}})^2 + \lg{\sqrt{2}} \times \lg{5} + \sqrt{(\lg{\sqrt{2}})^2 - \lg{2} + 1}$.

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