26953

My solution:

You will always end up with a multiple of three when you have chosen three numbers from the bags

The numbers in each bag are all a multiple of three plus one
Therefore if you choose three numbers there are three 'plus one's
(+1+1+1)
This equals another three
A multiple of three plus three is (obviously) another multiple of three

Here is a kind of diagram to explain again in case you don't yet understand:

These are the choices of numbers

1, 4, 7, 10

1=0+1
4=3+1
7=6+1
10=9+1

If you choose any three of these numbers (even allowing repeats) you will always end up with a multiple of three
As I said before
+1+1+1=3
Which then equals a multiple of 3