Trig Probability

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 $p+q$ ?