
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
If for any $x \in D$, it holds that $g(x) \leqslant f(x) \leqslant h(x)$, then the function $f(x)$ is considered an "arbitrary function" from $g(x)$ to $h(x)$ on the interval $D$. Given the functions $f(x)=kx$, $g(x)=x^{2}-2x$, $h(x)=(x+1)(\ln x+1)$, and that $f(x)$ is the "arbitrary function" from $g(x)$ to $h(x)$ on the interval $[1,e]$, find the range of the real number $k$.

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