Return values with matching conditions in r

I would like to return values with matching conditions in another column based on a cut score criterion. If the cut scores are not available in the variable, I would like to grab closest larger value. Here is a snapshot of dataset:

ids <- c(1,2,3,4,5,6,7,8,9,10)
scores.a <- c(512,531,541,555,562,565,570,572,573,588)
scores.b <- c(12,13,14,15,16,17,18,19,20,21)
data <- data.frame(ids, scores.a, scores.b)
> data
   ids scores.a scores.b
1    1      512       12
2    2      531       13
3    3      541       14
4    4      555       15
5    5      562       16
6    6      565       17
7    7      570       18
8    8      572       19
9    9      573       20
10  10      588       21

cuts <- c(531, 560, 571)

I would like to grab score.b value corresponding to the first cut score, which is 13. Then, grab score.b value corresponding to the second cut (560) score but it is not in the score.a, so I would like to get the score.a value 562 (closest to 560), and the corresponding value would be 16. Lastly, for the third cut score (571), I would like to get 19 which is the corresponding value of the closest value (572) to the third cut score.

Here is what I would like to get.

       scores.b
cut.1  13
cut.2  16
cut.3  19

Any thoughts? Thanks