4 integers between 100 and 999, inclusive, which equal the sum of the cubes of their digits - SMALL HINT

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?


Small Hint

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.


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