
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 $f(x) = x^3 - ax - 1$,
(1) If $a = 3$, determine the intervals where $f(x)$ is monotonic;
(2) If $f(x)$ is monotonically increasing on the set of real numbers $\mathbb{R}$, find the range of the real number $a$;
(3) Does there exist a real number $a$ such that $f(x)$ is monotonically decreasing on the interval $(-1, 1)$? If it exists, find the range of $a$; if it does not exist, explain why.

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