
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 $\left[\frac{1}{e}, e\right]$.

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

(Ⅲ) When $-1 < a < 0$, 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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