
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
Let $f(x)$ be a function defined on $\mathbb{R}$ that is never zero. For any real numbers $x, y \in \mathbb{R}$, it satisfies $f(x) \cdot f(y) = f(x+y)$. If $a_1 = \frac{1}{2}$ and $a_n = f(n)$ for $n \in \mathbb{N}^{*}$, determine the value range of the sum of the first $n$ terms $S_n$ of the sequence $\{a_n\}$.

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