
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
A sequence $\{a_n\}$ is defined such that the "harmonic mean" of its first $n$ terms is $\frac{1}{n+2}$.

1. Find the general formula for the $n$-th term of the sequence $\{a_n\}$.
2. If $C_n = \frac{a_n}{3^n}$, find the sum of the first $n$ terms of the sequence $\{C_n\}$, denoted by $S_n$.

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