Hey I think you should consider score verification. The CB website says that they’ll score your answer sheet by hand instead of machine scanning if you order score verification for your SAT. Also, there might be a possibility that you gridded incorrectly entire sections (CB says if they discover an obvious mistake in gridding–e.g. grid section 4 answers on section 5 or vice versa-- they’ll rescore the answer sheet as if you hadn’t gridded your answers on the wrong section).
For a native-speaker student who gets A’s and B’s, I don’t think an 1360 SAT score is normal even if it’s the first time you take it. I myself is a non-native speaker (I don’t live in US and don’t speak English at home) and got a 2250 the first time I take it–Math 800, writing 700, CR 750. Now I ended up with a 2310. So, your score seems to be an aberration, and if score verification can’t save you, then you must study for the SAT in a more strategic way.
I personally don’t think prep courses will necessarily help (I was an exchange student in sophomore year and went to the US to study for half a year, during which I took the ACT and its prep course, and I thought the course was crap). You know, in the US, SAT question-and-answer services are provided thrice a year: Oct, Jan, and May. CB releases these tests, and these are actual SAT reasoning tests that were administered, FROM THE MAKER OF THE TEST, and people have access to them. Don’t try Princeton, Kaplan or other guidebooks because there are subtle, if not lethal, differences between the logics of the maker of these guidebooks and the actual thinking of the maker of SAT. See if you can find those valuable released exams, and if you can, do practice. Answer explanations are usually appended, so you can get a glimpse of how the test makers themselves solve the questions on the test.
Mathematics sections are the easiest, but you need to be EXTREMELY careful because for an 800 you have to get everything correct. Critical reading sections are the most interesting, but sentence completion can be frustrating because of its requirement for large vocabulary. Try Barron’s essential word list or Merriam-Webster’s vocabulary builder if you need to quickly build up your vocab. Writing sections are hugely imbalanced. Although CB says the essay weighs approximately 30%, the writing multiple choice pretty much dominates the entire writing part (much more than its allocated 70%). So, to secure a 750+ writing score, you have to get almost every writing multiple choice question right (miss no more than one), although a few exceptions to this general rule can be found, including the May 2013 SAT administration.
If you struggle with Critical Reading passages and their following multiple choice questions, try to look for the “old” SAT verbal sections—when SAT didn’t yet include a writing part. Although these SAT tests date back to more than 15 years ago, their verbal section passages are incredibly good practice.
Increasing your SAT score by at least 800 points isn’t impossible. Be confident, and best of luck!