
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 all complex numbers with a modulus equal to 1). Suppose \( f \) is a mapping from \( S \) to \( S \) defined by:
$$
\begin{array}{l}
f^{(1)}(z) = f(z), \quad f^{(2)}(z) = f(f^{(1)}(z)), \ldots, \\
f^{(k+1)}(z) = f(f^{(k)}(z)), \quad k = 2,3, \ldots.
\end{array}
$$

If \( c \in S \) is such that
$$
f^{(1)}(c) \neq c, \ldots, f^{(n-1)}(c) \neq c, \quad f^{(n)}(c) = c,
$$

then \( c \) is called a period \( n \) point of \( f \).

Given a natural number \( m \) greater than 1, \( f \) is defined as follows:
$$
f(z) = z^{m}, \quad z \in S.
$$

Find the total number of period 1989 points for \( f \).

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