Since $a,b,c$ all have to be natural numbers, $a=1$ because it must be less than $\frac{10}{7}$, but also an integer greater than $0$. Furthermore, we now have 1b+1c=37 cbc+1=37. Therefore, we find that $c=3$ and $b=2$. Thus, our answer is $(a,b,c) = \boxed{(1,2,3)}.$
