
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
Find the smallest positive integer  $k$  with the following property: if each cell of a  $100\times 100$  grid is dyed with one color and the number of cells of each color is not more than  $104$ , then there is a  $k\times1$  or  $1\times k$  rectangle that contains cells of at least three different colors.

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