
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 least positive integer \( N \) with the following property: If all lattice points in \([1,3] \times [1,7] \times [1,N]\) are colored either black or white, then there exists a rectangular prism, whose faces are parallel to the \( xy \), \( xz \), and \( yz \) planes, and whose eight vertices are all colored in the same color.

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