Tantalizing Trigonometric Equations

Algebra Level 5

For a parameter \(a,\) suppose \(u\) is the smallest positive real number such that \[100\cos ^2 u -100\sin ^2 u =a+25\] and \(v\) is the largest negative real number such that \[25\sqrt{3}\sin v-75\cos v =a\] For how many integer values of \(a\) is \(u+v\) non-negative?

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