
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 $f\left(x\right)=|e^{x}-1|$, $x_{1} \lt 0$, $x_{2} \gt 0$, the graphs of the function $f\left(x\right)$ have two perpendicular tangents at points $A(x_{1}$, $f(x_{1}))$ and $B(x_{2}$, $f(x_{2}))$. These tangents intersect the $y$-axis at points $M$ and $N$ respectively. Find the range of $\frac{|AM|}{|BN|}$.

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