Let \( \hat{a} \) and \( \hat{b} \) be two unit vectors such that \( \left|(\hat{a}+\hat{b})+2(\hat{a}\times \hat{b})\right|=2 \). If \( \theta \in (0,\pi) \) is the angle between \( \hat{a} \) and \( \hat{b} \), then among the statements:
(S1): \( 2\left|\hat{a}\times \hat{b}\right|=\left|\hat{a}-\hat{b}\right| \)
(S2): The projection of \( \hat{a} \) on \( (\hat{a}+\hat{b}) \) is \( \frac{1}{2} \)
