A set of numbers contains 7 integers and has an average (arithmetic mean) value as well as a median value of 23. If the largest value is equal to 15 more than 4 times the smallest number, what is the largest possible range for the numbers in the set?

33
35
38
48
75
OA is 48

Is it a mistake to assume evenly spaced set of numbers for the sake of convenient calculation?
I did so and got the right answer. Here is my reasoning:
For and evenly spaced set of n numbers, the (T1+Tn)/2 = Mean, thus it will be equal to Median.
T1: 1st term of the set
Tn: nth term of the set
n is odd.

Therefore, (T1+Tn)/2 = 23 .... (i)
Now, let me plug in the ranges provided in the answer choices.
Tn = T1+ 75
From equation i,
T1+T1+75 = 23x2 = 46
T1 = 29/2, which is not possible as the set in question is a set of integers.

Similarly, if I plug in 48, the final equation comes out to be 2T1 = -2, T1 = -1, which is possible, -1 being an integer. The target question does not mention that the integers should be positive.
Also, 48 is the largest among the remaining options. So, my answer is 48.

Experts, I don't understand if this approach is correct.
The cross verification does not work.
"Largest value is equal to 15 more than 4 times the smallest number"
T1 = -1
T7 = -1+48 = 47
T7 = 4(-1) + 15 = 11 (It does not satisfy the target question)

Thanks