A tennis player computed her “win ratio” by dividing the number of matches she has won by the total number of matches she has played. At the start of a weekend, her win ratio is exactly 0.500. During the weekend, she plays four matches, winning 3 and losing 1. At the end of the weekend, her win ratio is greater than 0.503. What is the largest number of matches that she could have won before the weekend began?