
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 $f(x)=\frac{m}{x}+x\ln x$ $(m > 0)$, $g(x)=\ln x-2$:

1. When $m=1$, find the interval of $x$ for which $f(x)$ is increasing;

2. Let $h(x)=f(x)-xg(x)-\sqrt{2}$ $(x > 0)$. If the minimum value of the function $y=h(h(x))$ is $\frac{3\sqrt{2}}{2}$, find the value of $m$;

3. Suppose both $f(x)$ and $g(x)$ have the domain $[1,e]$, where $e$ is the base of natural logarithms. For any point $A$ on the graph of $f(x)$, there is a point $B$ on the graph of $g(x)$ such that $OA \perp OB$, with $O$ being the origin. Determine the range of values for $m$.

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