
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 their moduli equal to  $1$ ). 
We define function  $f:S\rightarrow S$  as follow:  $\forall z\in S$ , $ f^{(1)}(z)=f(z), f^{(2)}(z)=f(f(z)), \dots,$   $f^{(k)}(z)=f(f^{(k-1)}(z)) (k>1,k\in \mathbb{N}), \dots$ We call  $c$  an  $n$ -*period-point* of  $f$  if  $c$  ( $c\in S$ ) and  $n$  ( $n\in\mathbb{N}$ ) satisfy: $f^{(1)}(c) \not=c, f^{(2)}(c) \not=c, f^{(3)}(c) \not=c, \dots, f^{(n-1)}(c) \not=c, f^{(n)}(c)=c$ .
Suppose that  $f(z)=z^m$  ( $z\in S; m>1, m\in \mathbb{N}$ ), find the number of  $1989$ -*period-point* of  $f$ .

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