\[ \large \begin{cases}{ \color{blue}a \times \color{brown}b \times \color{green}c = \color{violet}{15}} \\ {\color{brown}b \times \color{green}c\times \color{red}d = \color{violet}{30}} \\ {\color{green}c\times \color{red}d \times \color{blue}a = \color{violet}{10}} \\ {\color{red}d \times \color{blue}a \times \color{brown}b = \color{violet}{6}}\end{cases} \]

Given \(a, b, c,\) and \(d\) are four distinct natural numbers that satisfy the system of equations above.

Determine the value of \(a+b+c+d\).

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