
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

The left side of the equation is nonnegative. Therefore, \( a x \geq 0 \). When \( a = 0 \), the equation has two roots, \(-1\) and \(1\). Thus, \( a = 0 \) does not satisfy the condition.

Consider the case \( a > 0 \). Then \( x \geq 0 \) and, therefore, \( |x|= x \). Construct the graph of the function \( y=|\ln x| \). The line \( y=a x \) must intersect this graph at three points. This is true if and only if the line passes inside the angle formed by the x-axis and the tangent \( y=a_{0} x \) to the graph of the function \( y=\ln x \) for \( x> 1 \).

Find \( a_{0} \). The abscissa of the tangent point satisfies the equations \( a_{0} x=\ln x, a_{0}=\frac{1}{x} \), yielding \( a_{0}=\frac{1}{e} \), and \( x=e \). Thus, \( 0 < a < \frac{1}{e} \).

The case \( a < 0 \) is symmetric, i.e., \( -\frac{1}{e} < a < 0 \).

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