
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 $f$ be a mapping from $S$ to $S$ defined by $f(z) = z^m$, where $m$ is a positive integer greater than 1. Define $f_1(z) = f(z)$ and $f_{k+1}(z) = f(f_k(z))$ for $k = 1, 2, 3, \cdots$. If there exists $c \in S$ and a positive integer $n$ such that $f_n(c) = c$ and $f_k(c) \neq c$ for all $k \in \mathbf{N}^+$ where $1 \leq k \leq n-1$, then $c$ is called an $n$-periodic point of $f$. 

Determine the number of $1989$-periodic points of $f$.

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