
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 $f(x) = \begin{cases} -\ln x, & 0 < x < 1 \\ \ln x, & x > 1 \end{cases}$, let lines $l$ and $m$ be the tangent lines to the graph of the function at points $M$ and $N$, respectively. It is known that $l$ and $m$ are perpendicular to each other and intersect the $y$-axis at points $A$ and $B$, respectively. Determine the range of possible values for the area of triangle $PAB$, where $P$ is any point on the graph of $y = f(x)$ for $x > 1$.

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