not impossible to cheat
bring up the start menu before begining the game, use the arrow keys to navigate that menu during the game until you can put your mouse on it wouthout touching a wall, then use the navigate the start menu by hovering the mouse untill you have it covering or next to exit, move mouse over start menu to exit, use arrow keys to close part of menu covering exit
TL;DR use start menu as bridge, have it open at all times, navigate it with arrows.