UA-UMO 2005 Problem 101

Let $0 \le \alpha, \beta, \gamma \le \frac{\pi}{2}$ satisfy the conditions

$\sin \alpha+\sin \beta+\sin \gamma=1, \sin \alpha \cos 2\alpha+\sin \beta\cos 2\beta+\sin \gamma \cos 2\gamma=-1.$

Find all possible values of $\sin^2 \alpha+\sin^2 \beta+\sin^2 \gamma$ .