
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 $f(x) = a\ln x - x^2$, for any two distinct real numbers $p$ and $q$ within the interval $(0,1)$, determine the range of the real number $a$ such that the inequality $\frac{f(p) - f(q)}{p - q} > 1$ always holds.

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