Step 1:
The average value of a function $f(x)$ over the interval $[a, b]$ is given by the formula:
$$\frac{1}{b-a}\int_{a}^{b}f(x)dx$$
In this case, our function is $f(x) = \frac{e^{3x}}{1+e^{6x}}$ and the interval is $\left[\frac{-\ln 3}{6}, 0\right]$.
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