
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
Let  $k$  and  $n$  be positive integers. Consider an array of  $2\left(2^n-1\right)$  rows by  $k$  columns. A  $2$ -coloring of the elements of the array is said to be *acceptable* if any two columns agree on less than  $2^n-1$  entries on the same row.

Given  $n$ , determine the maximum value of  $k$  for an acceptable  $2$ -coloring to exist.

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