TxO Math Bowl 2024 - Guts Contest - Set 8 Problem 2

Real numbers x, y, z satisfy the following conditions:

1. -1 \le x \le y \le z \le 1
2. x \ge \frac{1}{2}, or x + y \ge \frac{2}{3}, or x + y + z \ge \frac{3}{2}

Let \max\left(\min\left(\{y-x, z-y, 1-z\}\right)\right) = \frac{p}{q}, for relatively prime positive integers p and q. Compute p + q.