
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:  
(1) Given $a^{ \frac {1}{2}}+a^{- \frac {1}{2}}=3$, find $a+a^{-1}$;  
(2) $2(\lg \sqrt {2})^{2}+\lg \sqrt {2}\cdot \lg5+ \sqrt {(\lg \sqrt {2})^{2}-2\lg \sqrt {2}+1}$.

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