What's special about number 3?

mnp×npm×pmn=3mnp \Large \color{#3D99F6}{m}^{\color{#D61F06}{n}^{\color{#456461}{p}}} \times \color{#D61F06}{n}^{\color{#456461}{p}^{\color{#3D99F6}{m}}} \times \color{#456461}{p}^{\color{#3D99F6}{m}^{\color{#D61F06}{n}}} = \color{#EC7300}3\color{#3D99F6}{m}\color{#D61F06}{n}\color{#456461}{p}

Given that m,n,p\color{#3D99F6}{m}, \color{#D61F06}{n}, \color{#456461}{p} are positive integers that satisfy the equation above, what is the value of m+n+p \color{#3D99F6}{m} + \color{#D61F06}{n} + \color{#456461}{p} ?

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