Trig Probability

Algebra Level 4

Let $$m,n$$ be real numbers in the interval $$[-2,2]$$. Variables $$\alpha ,\beta$$ satisfy the system of equations: $\left\{\begin{array}{l} \sin \alpha + \cos \beta =m\\ \sin \beta + \cos \alpha = n \end{array}\right.$

Given that the numbers $$m,n$$ are picked at random, the probability that the system of equations has real solutions for $$\alpha , \beta$$ an be expressed as $$\dfrac{p\pi}{q}$$ for positive coprime integers $$p,q$$. What is $$p+q$$?

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