
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 any positive integer $n$, let $\langle n \rangle$ denote the floor of the logarithm base 2 of $n$, i.e. $\langle n \rangle = \lfloor \log_2{n} \rfloor$. Evaluate
\[
\sum_{n = 1}^\infty \frac{3^{\langle n \rangle} + 3^{-\langle n \rangle}}{3^n}.
\]

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