
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 $b$ satisfy the equation
\[\sqrt{\log{a}} + \sqrt{\log{b}} + \log \sqrt{a b^2} = 150\]
where all terms on the left side are positive integers, and log denotes the base 10 logarithm. What is the value of $a^2b^3$?
A) $10^{421}$
B) $10^{443}$
C) $10^{465}$
D) $10^{487}$

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