
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 $f(x)= \begin{cases} |\ln x|, & (0 < x\leqslant e^{3}) \\ e^{3}+3-x, & (x > e^{3})\end{cases}$, there exist $x\_1 < x\_2 < x\_3$ such that $f(x\_1)=f(x\_2)=f(x\_3)$. Find the maximum value of $\frac{f(x\_3)}{x\_2}$.

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