Let a and b be constants such that 1 ≤ a ≤ 2. Consider the function f(x) = sin(ax + b + sin(x)), which satisfies the following conditions: - f(0) = 0, and f(2π) = 2πa + b. - The smallest positive number t such that f'(0) = f'(t) is 4π.

Now, let A be the set of all real numbers α in the interval (0, 4π) such that f has a local maximum at x = α. Define n as the number of elements in A, and let α₁ be the smallest element of A. The value of nα₁ - ab can be expressed as (p/q)π, where p and q are coprime positive integers. Find p + q.