
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
Given that the sequence \(\left\{a_{n}\right\}\) is a geometric sequence with all positive terms, and \(a_{50}\) and \(a_{51}\) are two different solutions to the equation \(100 \lg^{2} x = \lg (100 x)\). Find the value of \(a_{1} a_{2} \cdots a_{100}\).

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