
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 $f(x)=\frac{1}{2}x^2-alnx-\frac{1}{2}$ where $a\in \mathbb{R}, a\neq 0$.  
$(1)$ Find the intervals where the function $f(x)$ is monotonic.  
$(2)$ If $f(x)\geq 0$ for all $x\in [1,+\infty)$, find the range of values for $a$.

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