
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 the function $f(x)=\sin 3x$, where $\frac{f(x_{i})}{x_{i}}=m$, $x_{i}\in[-2π,2π]$, $i=1$, $2$, $...n$, $n∈\mathbb{N}^{*}$, find the maximum value of $n$.

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