
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  $k,n$  be positive integers. There are  $kn$  points  $P_1, P_2, \cdots, P_{kn}$  on a circle. We can color each points with one of color  $ c_1, c_2, \cdots , c_k $ . In how many ways we can color the points satisfying the following conditions?

(a) Each color is used  $ n $  times.

(b)  $ \forall  i \not = j $ , if  $ P_a $  and  $ P_b $  is colored with color  $ c_i $  , and  $ P_c $  and  $ P_d $  is colored with color  $ c_j $ , then the segment  $ P_a P_b $  and segment  $ P_c P_d $  doesn't meet together.

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