Given that

\[ \frac{ \color{red}{a}}{\color{blue}{b}+\color{orange}{c}} + \frac{\color{blue}{b}} {\color{red}{a}+\color{orange}{c}} + \frac{ \color{orange}{c}}{\color{red}{a}+\color{blue}{b}} = 1, \]

find the value of

\[\large \frac{ \color{red}{a}^2}{\color{blue}{b}+\color{orange}{c}} + \frac{ \color{blue}{b}^2}{\color{red}{a}+\color{orange}{c}} + \frac{ \color{orange}{c}^2} { \color{red}{a}+\color{blue}{b}}. \]

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