TopsOJ Summer 2025 - Individual Round - Problem 12

Let a \uparrow b denote the power tower a^{a^{\ldots^a}}, where a is exponentiated b times in the tower (this includes the base). Given that x and y are positive integers such that \log_{2}\left( \log_{2 \uparrow y} 2^{2^{2^x}} \right)=496, compute the sum of all possible distinct x+y.