
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
Positive real numbers $a$ and $b$ have the property that
\[
2(\sqrt{\log{a}} + \sqrt{\log{b}}) + \log \sqrt{a} + \log \sqrt{b} = 108
\]
where all terms on the left are positive integers, and $\log$ denotes the base 10 logarithm. What is $ab$?
A) $10^{112}$ 
B) $10^{116}$ 
C) $10^{120}$ 
D) $10^{124}$

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