Christmas Contest - Individual Round - Problem 3

Geeks is playing a game of Entry Point with Victor, a fun multiplayer FPS game in Roblox. They are currently working together to complete a mission. If Geeks is talking about Bedwars, he gets distracted.

This leads Geeks to have a \dfrac{3}{4} chance of losing the game. If Geeks is not talking about Bedwars, and is focused, he has a \dfrac{1}{8} chance of losing the game. Victor has a \dfrac{1}{4} chance of talking about Bedwars and inadvertently distracting Geeks.

In addition, Victor has a \dfrac{1}{16} chance of losing the game. (Geeks and Victor are playing the same game, so if one of them lose, they both lose. In addition, the only 2 possible results of the game are either a win or a loss).

Let \dfrac{m}{n} denote the probability of Geeks and Victor winning the game where \gcd(m, n) = 1 and m, n \in \mathbb{Z}^+. Compute m + n.