
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 $f(x) = \begin{cases} \ln(1-x), & x < 1 \\ \frac{2}{x-1}, & x > 1 \end{cases}$, and $g(x) = \frac{k}{x^{2}}$ ($k > 0$), for any $p \in (1, +\infty)$, there always exist real numbers $m$, $n$ satisfying $m < 0 < n < p$, such that $f(p) = f(m) = g(n)$. The maximum value of the integer $k$ is \_\_\_\_\_\_.

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