
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
Let $a$ be the largest positive root of the equation $x^{3} - 3 x^{2} + 1 = 0$. Among the 2006 numbers $\left[a^{n}\right] (1 \leq n \leq 2006)$, how many of them are divisible by 17?

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