
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 $f(x)=\frac{1}{2}x^{2}+(1-k)x-k\ln x$.  
$(1)$ Discuss the monotonicity of $f(x)$.  
$(2)$ If $k$ is a positive number and there exists $x_{0}$ such that $f(x_{0}) \lt \frac{3}{2}-k^{2}$, find the range of values for $k$.

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