
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
It is known that $\lg x_{1}, \lg x_{2}, \lg x_{3}, \lg x_{4}, \lg x_{5}$ are consecutive positive integers (either in ascending or descending order), and $\left(\lg x_{4}\right)^{2} < \lg x_{1} \cdot \lg x_{5}$. What is the minimum value of $x_{1}$?

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