
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 $f(x) = -x^3 + 1 + a$ $(\frac{1}{e} \leq x \leq e, e$ is the base of natural logarithms$)$ and the graph of $g(x) = 3\ln(x)$ have points that are symmetric about the $x$-axis, find the range of the real number $a$.

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