
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
Find the smallest real number \(\lambda\) such that there exists a sequence \(\{a_n\}\) with all terms greater than 1, and for any positive integer \(n\), the following inequality is satisfied:
\[
\prod_{i=1}^{n+1} a_i < a_n^\lambda.
\]

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