
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 $f(x)=a\ln(x+1)-x^{2}$, where $a$ is a real number, and any two real numbers $p$ and $q$ are chosen in the interval $(0,1)$ such that $p\neq q$. If the inequality $\frac{f(p+1)-f(q+1)}{p-q} \gt 2$ always holds, determine the range of the real number $a$.

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