A two$-$player game: Fix a positive integer n$.$ Write out the integers $1,2,3,...,$ n$.$ First player chooses even or odd. Second player begins by writing $1$ next to any of the numbers in the list; first player then writes $2$ next to any remaining number in the list; they continue alternating until they have reached n$.$ Count the number of numbers written next to the same number in the list. First player wins if this count is the same parity $($even or odd$)$ they chose before the game began $($otherwise$,$ the second player wins$).$ Play and analyze this game.