
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), and let \(f\) be a mapping from \(S\) to \(S\). For any \(z \in S\), define \(f^{(1)}(z) = f(z)\), \(f^{(2)}(z) = f(f(z))\), and so on. In general, \(f^{(k)}(z) = f(f^{(k-1)}(z))\), etc. If \(c \in S\) and a positive integer \(n\) such that \(f^{(1)}(c) \neq c, f^{(2)}(c) \neq c, \cdots, f^{(n-1)}(c) \neq c, f^{(n)}(c) = c\), we say that \(c\) is an \(n\)-periodic point of \(f\). Let \(m\) be a positive integer greater than 1, and \(f\) is defined by:
\[ f(z) = z^{m}, \quad z \in S \]
Find the number of 1989-periodic points of \(f\).

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