GAIC Math 2024 Problem 132

Anna, Carl take turns selecting numbers from the set $\{1,2, \cdots, p-1\}$ (where $p$ is a prime greater than 3). Anna goes first, and each number can only be selected once. Each number chosen by Anna is multiplied by the number Carl selects next. Carl wins if, after any round, the sum of all products computed so far is divisible by $p$. Anna wins if, after all numbers are chosen, Carl has not won. Which of the following options is correct?

A. Anna has a winning strategy. B. Carl has a winning strategy. C. Both players have no winning strategy.