
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 on the complex plane (i.e., the set of complex numbers with modulus equal to 1), and \( f \) be a mapping from \( S \) to \( S \). For any \( z \in S \), define

\[
f^{(1)}(z) = f(z),\quad f^{(2)}(z) = f(f(z)),\quad \cdots,\quad f^{(k)}(z) = \underbrace{f(f(\cdots(f}_{k \text{ times}}(z) \cdots))),
\]

If \( c \in S \) satisfies

\[
f^{(1)}(c) \neq c,\quad f^{(2)}(c) \neq c,\quad \cdots,\quad f^{(n-1)}(c) \neq c,\quad f^{(n)}(c) = c,
\]

we call \( c \) an \( n \)-periodic point of \( f \).

Let \( m \) be a natural number greater than 1 and define \( f \) as follows:

\[
f(z) = z^{m} \quad (z \in S).
\]

Find the total number of 1989-periodic points of \( f \).

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