OPM 2024 Problem 6

Alexandre and Bernado are playing the following game. At the beginning, there are $n$ balls in a bag. At first turn, Alexandre can take one ball from the bag; at second turn, Bernado can take one or two balls from the bag, and so on. So they take turns and in $k$ turn, they can take a number of balls from $1$ to $k$ . Wins the one who makes the bag empty.

For each value of $n$ , find who has the winning strategy.