
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's a tape with  $n^2$  cells labeled by  $1,2,\ldots,n^2$ . Suppose that  $x,y$  are two distinct positive integers less than or equal to  $n$ . We want to color the cells of the tape such that any two cells with label difference of  $x$  or  $y$  have different colors. Find the minimum number of colors needed to do so.

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