ABC's are hauntin' you

Algebra Level 4

If a+b+c=0,\Large \color{#D61F06}{a}+\color{#3D99F6}{b}+\color{#20A900}{c} = 0, then  a5+b5+c5ab+bc+ca=Kabc.\ \Large \dfrac{\color{#D61F06}{a}^5 + \color{#3D99F6}{b}^5 + \color{#20A900}{c} ^5}{\color{#D61F06}{a}\color{#3D99F6}{b} + \color{#3D99F6}{b}\color{#20A900}{c} +\color{#20A900}{c} \color{#D61F06}{a}}=K\cdot \color{#D61F06}{a}\color{#3D99F6}{b}\color{#20A900}{c}. K = ?\Large K \ = \ ?

Note: Assume that a,b,ca,b,c are not all 0.

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