
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
For any positive integer \( q_0 \), consider the sequence defined by
$$
q_i = \left(q_{i-1} - 1\right)^3 + 3 \quad (i = 1, 2, \cdots, n)
$$
If every \( q_i (i = 1, 2, \cdots, n) \) is a power of a prime number, find the largest possible value of \( n \).

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