Challenge · Superchallenge problem

Let $f(x) = \dfrac{4^x}{4^x + 2}$. Compute $f\!\left(\tfrac{1}{100}\right) + f\!\left(\tfrac{2}{100}\right) + \cdots + f\!\left(\tfrac{99}{100}\right)$. The sum is $\tfrac{m}{n}$ in lowest terms; find $m + n$.