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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