Solve the following system of equations in real numbers: $$\left\{\begin{array}{l}x^2=y^2+z^2,\\x^{2024}=y^{2024}+z^{2024},\\x^{2025}=y^{2025}+z^{2025}.\end{array}\right.$$Proposed by Mykhailo Shtandenko, Anton Trygub, Bogdan Rublov
Source: Kyiv City MO 2024 Round 2, Problem 11.1
Tags: algebra, system of equations
Solve the following system of equations in real numbers: $$\left\{\begin{array}{l}x^2=y^2+z^2,\\x^{2024}=y^{2024}+z^{2024},\\x^{2025}=y^{2025}+z^{2025}.\end{array}\right.$$Proposed by Mykhailo Shtandenko, Anton Trygub, Bogdan Rublov