
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
Each of the positive integers  $1,2,\dots,n$  is colored in one of the colors red, blue or yellow regarding the following rules:
(1) A Number  $x$  and the smallest number larger than  $x$  colored in the same color as  $x$  always have different parities.
(2) If all colors are used in a coloring, then there is exactly one color, such that the smallest number in that color is even.
Find the number of possible colorings.

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