
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 \( S \) be the unit circle in the complex plane (i.e., the set of complex numbers with modulus equal to 1). Let \( f \) be a mapping from \( S \) to \( S \). For any \( z \in S \), define \( f^{(1)}(z) = f(z), f^{(2)}(z) = f(f(z)), \ldots, f^{(k)}(z) = f(f^{(k-1)}(z)) \).

If \( c \in S \) and a natural number \( n \) such that \( f^{(1)}(c) \neq c, f^{(2)}(c) \neq c, \ldots, f^{(n-1)}(c) \neq c, f^{(n)}(c) = c \), then \( c \) is called an \( n \)-periodic point of \( f \).

Let \( m \) be a natural number greater than 1, and \( f \) be defined as \( f(z) = z^m \) for \( z \in S \). Calculate the number of 1989-periodic points of \( f \).

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