Joan and Dick spent 2 weeks (14 nights) touring four cities on the East Coast: Boston, New York, Philadelphia, and Washington. They paid $240, $450, $170, and $100 per night for lodging in each city, respectively, and their total hotel bill came to $4200. The number of days they spent in New York was the same as the total number of days they spent in Boston and Washington, and the couple spent 3 times as many days in New York as they did in Philadelphia. How many days did Joan and Dick stay in each city? (Let x, y, z, and w denote the numbers of days spent in Boston, New York, Philadelphia, and Washington, D.C., respectively.)