\[ \Large \color{blue}{m}^{\color{red}{n}^{\color{darkgreen}{p}}} \times \color{red}{n}^{\color{darkgreen}{p}^{\color{blue}{m}}} \times \color{darkgreen}{p}^{\color{blue}{m}^{\color{red}{n}}} = \color{orange}3\color{blue}{m}\color{red}{n}\color{darkgreen}{p}\]

Given that \(\color{blue}{m}, \color{red}{n}, \color{darkgreen}{p} \) are positive integers that satisfy the equation above, what is the value of \( \color{blue}{m} + \color{red}{n} + \color{darkgreen}{p} \)?

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