Sarah had a collection of stamps. She decided to give some away over three weeks. In the first week, she gave away $\frac{1}{2}$ of her stamps plus 5 stamps. In the second week, she gave away $\frac{1}{3}$ of the remaining stamps plus 10 stamps. In the third week, she gave away $\frac{1}{4}$ of the remaining stamps plus 15 stamps. After the third week, she had 60 stamps left. How many stamps did Sarah have initially?