
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 $f(x)=\ln x+ax^2+2x+1$, for any two different positive real numbers $x_1$ and $x_2$, the inequality $\frac{f(x_1)-f(x_2)}{x_1-x_2}\geqslant 4$ always holds, determine the range of values for the real number $a$.

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