
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
In a  $25 \times n$  grid, each square is colored with a color chosen among  $8$  different colors. Let  $n$  be as minimal as possible such that, independently from the coloration used, it is always possible to select  $4$  coloumns and  $4$  rows such that the  $16$  squares of the interesections are all of the same color. Find the remainder when  $n$  is divided by  $1000$ . 

*Proposed by **FedeX333X***

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