A positive integer \( n \) satisfies the following conditions: The number \( n \) has exactly \( 60 \) divisors: \( 1 = a_1 < a_2 < \cdots < a_{60} = n \); The number \( n+1 \) also has exactly \( 60 \) divisors: \( 1 = b_1 < b_2 < \cdots < b_{60} = n+1 \). Let \( k \) be the number of indices \( i \) such that \( a_i < b_i \). Find all possible values of \( k \). Note: Such numbers exist, for example, the numbers \( 4388175 \) and \( 4388176 \) both have \( 60 \) divisors. Proposed by Anton Trygub