
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
There are $n$  balls numbered  $1,2,\cdots,n$ , respectively. They are painted with  $4$  colours, red, yellow, blue, and green, according to the following rules:
First, randomly line them on a circle.
Then let any three clockwise consecutive balls numbered  $i, j, k$ , in order.
1) If  $i>j>k$ , then the ball  $j$  is painted in red;
2) If  $i<j<k$ , then the ball  $j$  is painted in yellow;
3) If  $i<j, k<j$ , then the ball  $j$  is painted in blue;
4) If  $i>j, k>j$ , then the ball  $j$  is painted in green.
And now each permutation of the balls determine a painting method.
We call two painting methods distinct, if there exists a ball, which is painted with two different colours in that two methods.

Find out the number of all distinct painting methods.

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