Let ℕ = {1, 2, 3, . . .} be the set of all positive integers.
Find all functions f : ℕ → ℕ such that for any positive integers a and b, the following two conditions hold:
(1) f(ab) = f(a)f(b)
(2) at least two of the numbers f(a), f(b) and f(a + b) are equal.
Source: European Girls’ Mathematical Olympiad - 2022 - Day 1 - Problem 2
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