Holiday Contest 2025 - Individual Round - Problem 6

A spinner wheel is split into three sectors, each labeled with positive integers a, b, and c. The number on each sector is proportional to the probability the sector gets landed on when the wheel is spun. For example, a sector labeled with 6 is three times as likely to be landed on than a sector labeled with 2. When the wheel is spun, the expected value of the label of the sector the wheel lands on is 7. Compute the sum of all possible values of a+b+c.