
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 $f(x)=a\ln x+ \frac {1}{2}x^{2}+(a+1)x+1$.
$(1)$ When $a=-1$, find the interval of monotonic increase for the function $f(x)$;
$(2)$ If the function $f(x)$ is increasing on $(0,+\infty)$, find the range of the real number $a$;
$(3)$ If $a > 0$, and for any $x_{1}$, $x_{2}\in(0,+\infty)$, $x_{1}\neq x_{2}$, it always holds that $|f(x_{1})-f(x_{2})| > 2|x_{1}-x_{2}|$, find the minimum value of the real number $a$.

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