
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
Define a power cycle to be a set \( S \) consisting of the nonnegative integer powers of an integer \( a \), i.e. \( S = \{1, a, a^2, \ldots\} \) for some integer \( a \). What is the minimum number of power cycles required such that given any odd integer \( n \), there exists some integer \( k \) in one of the power cycles such that \( n \equiv k \pmod{1024} \)?

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