
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 $f(x)=alnx-(a+1)x+\frac{1}{2}x^2$.  
$(1)$ Find the intervals where $f(x)$ is monotonic.  
$(2)$ When $a \gt 0$, for any $x\in (1,+\infty)$, the inequality $f(x)≥x^a-e^x+\frac{1}{2}x^2-ax$ always holds. Find the range of real numbers for $a$.

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