
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
Factor the polynomial \(x^{12} - 1\) into its components where each component is a non-constant polynomial with real coefficients. Find the largest possible value of \(k\) such that 
\[ x^{12} - 1 = p_1(x) p_2(x) \dotsm p_k(x). \]

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