
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
Let lines $l_1$ and $l_2$ be the tangents at points $P_1$ and $P_2$ on the graph of the function $f(x)=\begin{cases}-\ln x, & 0 < x < 1 \\ \ln x, & x > 1\end{cases}$, respectively. If $l_1$ and $l_2$ intersect perpendicularly at point $P$, and $l_1$, $l_2$ intersect the $y$-axis at points $A$ and $B$, respectively, determine the range of the area of $\triangle PAB$.

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