Hello and thank you for your question,
Although it is going to be difficult to lose over 250 lbs with diet and exercise, it is doable. Whether or not you are going to be able to loose the weight depends on how determined you are and how long you can continue a diet for. I definitely think you should try diet and exercise before reverting to surgery (either gastric bypass or lap band).
The fact that you are young gives you an advantage and makes it easier to loose weight.
The 13 lbs that you lost in the first 8 days is mostly water weight but it is a great start.
You should not eat a vegetable diet ONLY, you need protein in your diet and you need healthy carbohydrates. If you stick to vegetables only, you will become malnourished and you will never achieve proper weight loss.
If you are very determined about trying to loose weight with diet/exercise, it is very important that you speak to a nutritionist who can help you plan out well balanced meals, not just vegetables alone.
I hope you find this information helpful. Please feel free to follow-up with any other questions you might have.
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Best wishes to you,