JBMO-SL 2020 C3

Alice and Bob play the following game: Alice begins by picking a natural number $n \geqslant 2$. Then, with Bob starting first, they alternately choose one number from the set $A=\{1,2, \ldots, n\}$ according to the following condition: The number chosen at each step should be distinct from all the already chosen numbers, and should differ by 1 from an already chosen number. (At the very first step Bob can choose any number he wants.) The game ends when all numbers from the set $A$ are chosen.

For example, if Alice picks $n=4$, then a valid game would be for Bob to choose 2, then Alice to choose 3, then Bob to choose 1, and then Alice to choose 4 .

Alice wins if the sum $S$ of all of the numbers that she has chosen is composite. Otherwise Bob wins. (In the above example $S=7$, so Bob wins.)

Decide which player has a winning strategy.