With five numbers there are 5 ways to group 4 of them together. If you take the averages of each group of 4 numbers and get...

3428.5
3425.75
3398
3120.25
342.5

What are the 5 numbers?

If the numbers are named x(1-5), and T = the sum of the five numbers, each of the average given is equal to 1/4 of (T less one of the searched numbers) :

T - x1 = 3428.5 * 4 = 13714
T - x2 = 3425.75 * 4 = 13703
T - x3 = 3398 * 4 = 13592
T - x4 = 3120.25 * 4 = 12481
T - x5 = 342.5 * 4 = 1370

When I sum these 5 lines :

5*T-(x1+x2+x3+x4+x5)=4*T=54860 => T=13715

Then, from the five lines :

x1=1
x2=12
x3=123
x4=1234
x5=12345