Neither
Rapid weight loss is the goal of the fat fast not permanent.
The life change that you have to go through to get weight to stay off is the entire plan and that means making the changes for the rest of your life by learning what foods you can and cant eat.
You can stay with induction as long as you want to lose weight and then move on to the other phases of Atkins. You can start with the fat fast and then move onto induction -12 days is the maximum of the fat fast where you limit your calories to 1000 for 4 days then 1200 for a week. The menu is more varied than the egg fast, but it should give the same results.
Only if you are resistant to weight loss should you try this.