
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 $f(x) = (a+1)\ln x + ax^{2} + 1$, where the constant $a \in \mathbb{R}$.  
$(1)$ When $a=0$, find the zeros of $f(x)$.  
$(2)$ Discuss the monotonicity of $f(x)$.  
$(3)$ Let $a$ be a real number such that $a \lt -1$. If for any $x_{1}, x_{2} \in (0, +\infty)$, the inequality $|f(x_{1}) - f(x_{2})| \geq 4|x_{2} - x_{1}|$ holds, find the range of values for the real number $a$.

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