TxO Math Bowl 2024 - Individuals B - Problem 4

Let a, b, c, d be distinct integers in the set \{1, 2, \dots, 9, 10\}. They satisfy the following equations:

\begin{cases} a + c + d = b \\ c = 2a \\ a \equiv 0 \pmod 3 \end{cases}

If the sum of all rational roots of the equation ax^3 + bx^2 + cx + d = 0 can be expressed as -\frac{m}{n}, where m and n are relatively prime positive integers, find the value of m + n.