
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 $f(x)=\frac{a}{x+1}+\ln x$. If for any $x_{1}$, $x_{2}\in (0,2]$, and $x_{1}\neq x_{2}$, we have $\frac{f({x}_{2})-f({x}_{1})}{{x}_{2}-{x}_{1}} > -1$, determine the range of real numbers $a$.

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