
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 a function $f(x)$ that satisfies $f(x+y)=f(x) \times f(y)$, and $f(1)= \frac{1}{2}$,

$(1)$ When $n \in \mathbb{N}^*$, find the expression for $f(n)$.

$(2)$ Let $b_n=(9-n) \frac{f(n+1)}{f(n)}$, where $n \in \mathbb{N}^*$, and $S_n$ be the sum of the first $n$ terms of $\{b_n\}$. When $S_n$ is maximized, find the value of $n$.

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