\begin{document}$ p=1/2 $\end{document} was primarily selected for discussion. To understand the impact of the hairy parameter on black holes, we have studied the event horizon radius \begin{document}$ r_{\rm h} $\end{document}, photon sphere radius \begin{document}$ r_{\rm ph} $\end{document}, and radius of the innermost stable circular orbit \begin{document}$ r_{\rm isco} $\end{document} of this hairy black hole. Subsequently, we utilize the backward ray-tracing method to analyze the geodesics of photons around this black hole and discuss the influence of the hairy parameter on the photon geodesics. In addition, we calculate the distinctive shadow and photon ring structures of the black hole illuminated by a static thin accretion disk using three toy-model emission functions, and we briefly compare the optical appearance of the black hole in this nonlinear case (\begin{document}$ p=1/2 $\end{document}) with that in the standard Yang-Mills case (\begin{document}$ p=1 $\end{document}). The research results show that as the hairy parameter gradually increases, \begin{document}$ r_{\rm h} $\end{document}, \begin{document}$ r_{\rm ph} $\end{document}, \begin{document}$ r_{\rm isco} $\end{document}, and the critical impact parameter \begin{document}$ b_{\rm ph} $\end{document} of the black hole all exhibit a decreasing trend. Meanwhile, it also causes the area of the black hole shadow and the photon ring to decrease accordingly. Compared with the standard case (\begin{document}$ p=1 $\end{document}), the nonlinear Einstein-Maxwell Yang-Mills black hole exhibits a larger shadow radius and brighter ring radius but with an overall dimmer brightness relative to the standard scenario, demonstrating observable differences. Consequently, for the static and spherically symmetric Einstein-Maxwell power-Yang-Mills hairy black hole, no degeneracy occurs in the photon ring and shadow. Theoretically, it can reflect different black hole solutions and thus verify the Yang-Mills hair."> Testing Einstein-Maxwell Power-Yang-Mills hair via black hole photon rings -
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