Do not push
at any point while you are playing. This is the only thing that will stop you from getting this achievement.
You CAN press
to open up your menu. This let's you "pause" without violating the achievement requirements.
When you open the menu you can level up, heal, rest, and equip your heroes. So if you're in a pinch you can press
to open this menu, "pause" and heal your heroes or rest them to re-use their abilities.
The only things you can't do are assign hero locations, power/unpower rooms and build modules like you could if you pressed