
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
On a rectangular board with  $m$  rows and  $n$  columns, where  $m\leq n$ , some squares are coloured black in a way that no two rows are alike. Find the biggest integer  $k$  such that for every possible colouring to start with one can always color  $k$  columns entirely red in such a way that no two rows are still alike.

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