
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 $f(x)=a\ln x+\frac{a+1}{2}x^2+1$.

(I) When $a=-\frac{1}{2}$, find the maximum and minimum values of $f(x)$ in the interval $[\frac{1}{e}, e]$.

(II) Discuss the monotonicity of the function $f(x)$.

(III) When $-1 < a < 0$, the inequality $f(x) > 1+\frac{a}{2}\ln (-a)$ always holds. Find the range of $a$.

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