
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 \( q_{i} = \left( q_{i-1} - 1 \right)^3 + 3 \) for \( i = 1, 2, \cdots, n \). If each \( q_{i} (i = 1, 2, \cdots, n) \) is a prime number, find the largest possible value of \( n \).

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