
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 that lines $l\_1$ and $l\_2$ are tangent lines to the graph of the function $f(x) = \begin{cases} -\ln x, & 0 < x < 1\\ \ln x, & x > 1\end{cases}$ at points $P\_1$ and $P\_2$, respectively. Lines $l\_1$ and $l\_2$ intersect perpendicularly at point $P$, and intersect the $y$-axis at points $A$ and $B$, respectively. Determine the range of possible values for the area of triangle $PAB$.

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