
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 $f(x)=a\ln x+ \frac {a+1}{2}x^{2}+1$.
$(1)$ When $a=- \frac {1}{2}$, find the maximum and minimum values of $f(x)$ in the interval $[\frac {1}{e},e]$;
$(2)$ Discuss the monotonicity of the function $f(x)$;
$(3)$ When $-1 < a < 0$, for any $x > 0$, it always holds that $f(x) > 1+ \frac {a}{2}\ln (-a)$. Find the range of $a$.

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