Three numbers are chosen independently at random, one from each of the three intervals $[0, L_i](i=1, 2,3)$. If the distribution of each random number is uniform with respect to length in the interval it is chosen from, determine the expected value of the smallest of the three numbers chosen. The original answer is in the format $\frac{a}{b} L_{1}^{2} L_{2} L_{3}-\frac{c}{d} L_{1}^{3}\left(L_{2}+L_{3}\right)+\frac{e}{f} L_{1}^{4}$, please give the value of a+b+c+d+e+f.

Let's think step by step and output the final answer within \boxed{}.
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