Proving that if $a^n+b^n \mid a^m+b^m$, then $n \mid m$

Question

I came across this question in the book An Excursion in Mathematics:

Let $a,b,m,n \in \Bbb{N}, a\gt 1$ and $\gcd(a,b)=1$. Prove that if $a^n+b^n \mid a^m+b^m$, then $n \mid m$.

I have made numerous attempts:

1. Tried modifying the statement $k(a^n+b^n) = a^m+b^m$.
2. Played around with the expanded forms of $(a+b)^n$ and $(a+b)^m$.
3. Tried finding $\gcd(a+b,a^m+b^m)$.

But all to no avail. Can I have a hint as to how to continue on this problem?