The condominium of numbers!

23456789\huge\color{#624F41}2^{\color{#D61F06}3^{\color{#EC7300}4^{\color{#20A900}5^{\color{teal}6^{\color{magenta}7^{\color{#302B94}8^{\color{#3D99F6}9}}}}}}}

Find the last three digits of the number above.

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