
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 $f(x)=\frac{x^{2}}{2}-k\ln x$, where $k\in R$.
(1) Find the monotonicity of $f(x)$;
(2) Determine whether the equation $f(x)=0$ has solutions in the interval $(1, \sqrt {e})$. If there are solutions, state the number of solutions and the basis; if there are no solutions, state the reason.

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