To find the inverse function

Question

I have the following function, $$f(m) = a\left( {-m \ \exp(-2m) \over 2} - {\exp(-2m)\over 4} + {1 \over 4} \right) $$ where $m \ge 0$, $<m>$ = 1 and $a$ is just a constant

and I want to re-arrange this function such that, $$m = F(f(m))$$ Where $F$ is a function of $f(m)$.
I have made some progress in solving the above problem but I am stuck at a step. I have types my progress below. $$f(m) = a\left( {-m \ \exp(-2m) \over 2} - {\exp(-2m)\over 4} + {1 \over 4} \right)$$ $${4 \times f(m) \over a} - 1 = \left( {-2 \times m \ \exp(-2m) } - {exp(-2m)} \right)$$ $${4 \times f(m) \over a} - 1 = \left( {-2 \times m \ } - 1 \right) \times \exp(-2m)$$ I did try taking a log but it did not take me anywhere. Can someone help me? Thank you!