According to Kaplan, you should refer directly back to the prompt only to clarify any specific words in the statement that need clarifying before you discuss the meaning of the statement as a whole. When explaining what the statement means, you should only be referring to your thesis statement explaining when and why the prompt holds true. Kaplan encourages their students to focus on making the bulk of their explanation oriented toward examples to support the thesis, so you will mainly be using the prompt to formulate your thesis.
Thesis, Antithesis, and Synthesis are words used by the Princeton Review (and maybe others) to describe the 3 objectives of the essay.
Thesis is when you explain what the statement means; when and why it is valid.
Antithesis is when you describe a case in which the prompt is not true. Synthesis is when you explain the circumstances/factors that determine when the prompt is and is not true.