There are four integers between 100 and 999, inclusive, which equal the sums of the cubes of their digits. Three of them are 153, 370, and 407. For example, 153 = 1³ + 5³ + 3³. What is the fourth such integer?
Some 3-digit integers can be ruled out quickly. For instance, no 3-digit integer with 2 nines in it could work, since 9³ = 729, and 2 times that would already far exceed the maximum total of 999.