
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 $f(x)=\frac{1}{2}|{{e^x}-1}|$. If there exist $x_{1}$ and $x_{2}\in \left(-a,a\right)(x_{1} \lt x_{2}$,$a \gt 0)$, such that the tangents of the curve $y=f\left(x\right)$ at $x=x_{1}$ and $x=x_{2}$ are perpendicular to each other, then the range of real number $a$ is ______.

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