You are asked to implement the core update logic for a Particle Swarm Optimization (PSO) algorithm. The goal is to find the minimum of the function f(x) = x² + 5 * sin(x) on the interval [-10, 10].

All the parameters (like inertia w, cognitive c1, and social c2) and initial lists (positions, velocities, pbest_positions, pbest_values, gbest_position, gbest_value) are already set up for you.

Your task is to fill in the main iterative process:

1. Inside the inner for loop (for each particle i), you must first update the particle's velocity. Use the full PSO velocity update equation: w * velocities[i] + c1 * r1 * (pbest_positions[i] - positions[i]) + c2 * r2 * (gbest_position - positions[i]).
2. Update the particle's position by adding its new velocities[i] to positions[i].
3. After calculating the new value at positions[i], check if this new value is better than the particle's personal best (pbest_values[i]). If it is, you must update both pbest_positions[i] and pbest_values[i].
4. After the inner loop (which updates all particles), check if the new best personal value (which you find and store in min_value) is better than the current global best (gbest_value). If it is, update both gbest_value and gbest_position.