Advanced Nuclear Energy Technologies Act This bill directs the Department of Energy (DOE) to advance the research and development of domestic advanced, affordable, and clean nuclear energy. Specifically, DOE must (1) complete at least two advanced nuclear reactor demonstration projects by December 31, 2025, and (2) demonstrate at least two and not more than four additional operational advanced reactor designs by December 31, 2035. DOE must establish goals for research relating to advanced nuclear reactors, including by coordinating with members of private industry to advance the demonstration of various designs of advanced nuclear reactors. DOE must also develop and update biennially a 10-year strategic plan for its Office of Nuclear Energy.