
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 $f(x)=x^{2}\ln x+x$ and the function $g(x)=\frac{1}{2}k{x}^{2}-x+3$, if there exists a real number $x_{0}$ such that $f(x_{0})={g'}(x_{0})$, then the range of real number $k$ is ______.

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