One to rule three cubes

Let the equation \( 1 = (1+9m^3)^3 + (9m^4)^3 + (-9m^4-mx)^3 \) have solutions for all \( m \in \mathbb{N} \) and a certain \(x \in \mathbb{N} \).

What is the smallest positive integer \( x \) for this to be true?

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