Caitlin and Donal play a game called Basketball Shoot-Out. The game consists of $10$ rounds. In each round, Caitlin and Donal both throw a ball simultaneously at each other's basket. If a player's ball falls into the basket, that player scores one point; otherwise, they score zero points. The scoreboard shows the complete sequence of points scored by each player in each of the $10$ rounds of the game. It turns out that Caitlin has scored at least as many points in total as Donal after every round of the game. Prove the number of possible scoreboards is divisible by $4$ but not by $8$.