
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 $a_{50}, a_{51}$ are the two distinct solutions of 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