
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 $f(x) = \ln x - \frac{m(x+n)}{x+1}$ (where $m > 0, n \in \mathbb{R}$) is not monotonically increasing on the interval $(0, +\infty)$. If the inequality $m - n > \lambda$ always holds, determine the range of the real number $\lambda$.

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