Importance of Base 10

\[\large{\begin{eqnarray} g(x) = &&(4a - 3d) x^5 + (4b - 3e) x^4 + (4c - 3f) x^3 \\ &+& (4d - 3a) x^2 + (4e - 3b) x + 4f - 3c \end{eqnarray}}\]

We define the function \(g(x) \) as above where \(a,b,c,d,e \) and \(f\) are single-digit natural numbers independent of \(x\). If \(g(10) = 0\), find the value of \(a+b+c+d+e+f\).

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