
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 are wise men standing in a line, each wearing either a black or white hat. Each wise man can see the hats of all those in front of them. Starting from the last person in line to the first, each wise man will call out a color (either white or black) along with a natural number of their choice. At the end, the number of wise men who correctly named the color of their hat will be counted, and the entire group will receive a salary bonus for that many days.

The wise men are allowed to agree on a strategy beforehand. They also know that exactly \( k \) of them are insane (but they do not know which ones). An insane wise man will call out a color and a number regardless of any agreements. 

What is the maximum number of days with a salary bonus that the wise men can guarantee the group, regardless of the positions of the insane wise men in the line?

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