
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
Given a complete bipartite graph on  $n,n$  vertices (call this  $K_{n,n}$ ), we colour all its edges with  $2$  colours , red and blue . What is the least value of  $n$  such that for any colouring of the edges of the graph , there will exist at least one monochromatic  $4$  cycle ?

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