
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
Let \( a, b \in \mathbf{R}_{+} \) and satisfy the equation:
\[
\sqrt{\lg a} + \sqrt{\lg b} + \lg \sqrt{a} + \lg \sqrt{b} = 100,
\]
where \( \sqrt{\lg a}, \sqrt{\lg b}, \lg \sqrt{a}, \lg \sqrt{b} \in \mathbf{Z}_{+} \). Find the value of \( ab \).

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