
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 $f(x) = e^x + \frac{1}{e^x}$ (where $e$ is the base of the natural logarithm), if the positive number $a$ satisfies: there exists $x_0 \in [1, +\infty)$, such that $f(x_0) < a(-x_0^3 + 3x_0)$, then the range of values for $a$ is _______.

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