
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
For a positive real number \( a \), the inequality 

\[
\frac{a^{1}}{\left(1+3^{1}\right)\left(1+3^{2}\right)}+\frac{a^{2}}{\left(1+3^{2}\right)\left(1+3^{3}\right)}+\cdots+\frac{a^{n}}{\left(1+3^{n}\right)\left(1+3^{n+1}\right)}<\frac{1}{8}
\]

holds for any positive integer \( n \). Find the range of values of \( a \).

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