TxO Math Bowl 2024 - Team Contest - Problem 13

Let a, b, c, d be non-negative reals such that a + b + c + d = 12. If the sum of the maximum and the minimum value of:

\sqrt{a+1} + \sqrt{2b+1} + \sqrt{3c+1} + \sqrt{6d+1}

can be expressed as p + \sqrt{q} + r\sqrt{s}, where p, q, r, and s are positive integers, and q and s are not divisible by the square of any prime, compute p + q + r + s.