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?