GCD ... period!

Number Theory Level 3

For \(m>1\), it can be proven that the integer sequence \(f_m(n) = \gcd(n+m,mn+1)\) has a fundamental period \(T_m.\) In other words, \[\forall n \in \mathbb{N}, \space f_m(n+T_m) = f_m(n).\] Find an expression for \(T_m\) in terms of \(m,\) and then give your answer as \(T_{12}.\)