
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 the function $f(x)$ is an odd function defined on $(-\infty, 0) \cup (0, +\infty)$, and when $x > 0$, $f(x) = e^{x} - ax + e^{3}$, if there exists an arithmetic sequence $x_{1}$, $x_{2}$, $x_{3}$, $x_{4}$ ($x_{1} < x_{2} < x_{3} < x_{4}$), and $x_{1} + x_{4} = 0$, such that the sequence $\{f(x_{n})\} (n=1,2,3,4)$ forms a geometric sequence, determine the minimum value of $a$.

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