The sequence \( (a_n)_{n\geq 1} \) of positive integers is such that \( a_1 = 1 \) and \( a_{m+n} \) divides \( a_m + a_n \) for any positive integers \( m \) and \( n \). a) Prove that if the sequence is unbounded, then \( a_n = n \) for all \( n \). b) Does there exist a non-constant bounded sequence with the above properties? (A sequence \( (a_n)_{n\geq 1} \) of positive integers is bounded if there exists a positive integer \( A \) such that \( a_n \leq A \) for all \( n \), and unbounded otherwise.)