
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
Find all values of \( a \) such that the roots \( x_1, x_2, x_3 \) of the polynomial \( x^3 - 6x^2 + ax + a \) satisfy the equation
\[
\left(x_1 - 3\right)^3 + \left(x_2 - 3\right)^3 + \left(x_3 - 3\right)^3 = 0.
\]
(1983 Austrian Mathematical Olympiad Problem)

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