Extremely Divisible Polynomial

Consider all degree 4 monic polynomials \( f(x) \) with complex coefficients which satisfy the condition that for all positive integers \(n\), \( f(x) \) divides \( f(x^n) \).

Over all such polynomials, what is the smallest possible positive integer value of \( f(-3) \)?

(As a followup, can you describe all polynomials such that \( f(x) \mid f(x^n) \) for all positive integers \(n\)?)

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