Each of the natural numbers from $1$ to $n$ is colored either red or blue, with each color being used at least once. It turns out that:
- Every red number is a sum of two distinct blue numbers; and
- Every blue number is a difference between two red numbers.

Determine the smallest possible value of $n$ for which such a coloring exists.

Let's think step by step and output the final answer within \boxed{}.
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