Now I Know My ABCs

Calculus Level 2

Given that a\color{#20A900}{a}, b\color{#D61F06}{b} and c\color{#3D99F6}{c} are nonnegative real numbers such that abc=bc,\Large \color{#20A900}{a}^{\color{#D61F06}{b}^{\color{#3D99F6}{c}}}=\color{#D61F06}{b}^{\color{#3D99F6}{c}}, find the maximum possible value of a\color{#20A900}{a}.

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