Tantalizing Trigonometric Equations

Algebra Level 5

For a parameter a,a, suppose uu is the smallest positive real number such that 100cos2u100sin2u=a+25100\cos ^2 u -100\sin ^2 u =a+25 and vv is the largest negative real number such that 253sinv75cosv=a25\sqrt{3}\sin v-75\cos v =a For how many integer values of aa is u+vu+v non-negative?

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