Move the star symbol into the top-right-corner of the union symbol

I want to move the * into the top-right-corner of the union symbol.

documentclass{article}
begin{document}
    $${bigcuplimits_{i=1}^{k}}^* A_i$$ 
end{document}

I don’t know how to move *. I try use silly way vspace and it doesn’t work.

documentclass{article}
begin{document}
    $${bigcuplimits_{i=1}^{k}}^{vspace{-0.2cm}*} A_i$$ 
end{document}

So, how to move down the * symbol?

documentclass{article}
usepackage{amsmath} % for "sideset" macro
begin{document}
[
sideset{}{^*}bigcup_{i=1}^{k} A_i qquad
sideset{}{^*}bigcup_{i=1}^{k} ! A_i
]
end{document}