
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 functions $f(x)=\frac{m}{x}+x\ln x (m > 0)$ and $g(x)=\ln x-2$.
1. Find the interval where function $f(x)$ is increasing when $m=1$.
2. If for any $x\_1\in[1,e]$, there always exists $x\_2\in[1,e]$ such that $\frac{f(x\_1)}{x\_1}\cdot \frac{g(x\_2)}{x\_2}=-1$, where $e$ is the base of the natural logarithm, find the range of values for the real number $m$.

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