Given $n \in \mathbb{N}$, define $ g_n(d) = f^{(d)}(n) - 667d. $ Routine computation yields the recurrence equation $ g_n(d+3) = 3g_n(d+2) - 6g_n(d+1) + 4g_n(d). $ The characteristic polynomial is $ x^3 - 3x^2 + 6x - 4 = (x-1)(x^2-2x+4). $ The two roots of $x^2-2x+4$ are $2e^{\pm i \pi/3}$.
It follows from the general theory of recurrence relations that for some real $A,B$ and angle $\theta$, $g_n(d) = A + B2^d \cos (d\pi/3 + \theta).$ The cosine part is a periodic function in $d$ attaining $6$ fixed values, at least one of them strictly positive and at least one strictly negative. In particular, if $B \neq 0$ then for some $e \in \{0,\ldots,5\}$ we have $C \triangleq B\cos (e\pi/3 + \theta) < 0.$ This implies that $f^{(6d+e)}(n) = g_n(6d+e) + 4002d + 667e = A + 2^d C + 4002d + 667e $ is eventually negative. This contradicts the premise that $f \geq 0$, and so $B = 0$. Substituting $d = 0$ in the formula for $g_n$, we deduce that $g_n(d) = A = n$ and so $ f^{(d)}(n) = n + 667d. $ In particular, $f(n) = n + 667$.