
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 $n$ houses in a row painted in $k$ different colors. For any given color, there exist 100 consecutive houses such that the number of houses of this color is strictly greater than the number of houses of any other color. What is the largest possible value of $k$ if: a) $n=404$? b) $n=406$?

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