
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
Given that the sum of the first $n$ terms, $S_{n}$, of a sequence ${a_{n}}$ satisfies the relationship $S_{n}=\frac{1}{2}-\frac{1}{2}a_{n}$.

1. Find the general formula for the $n$-th term of the sequence ${a_{n}}$.
2. Let $f(x)=\log_{3}{x}$ and $b_{n}=a_{n}f(a_{n})$. Find the sum of the first $n$ terms of the sequence ${b_{n}}$, denoted by $S_{n}$.

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