
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
rainbow is the name of a bird. this bird has  $n$  colors and it's colors in two consecutive days are not equal. there doesn't exist  $4$  days in this bird's life like  $i,j,k,l$  such that  $i<j<k<l$  and the bird has the same color in days  $i$  and  $k$  and the same color in days  $j$  and  $l$  different from the colors it has in days  $i$  and  $k$ . what is the maximum number of days rainbow can live in terms of  $n$ ?

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