已知两个不相等的非零向量$\stackrel{\rightharpoonup }{a},\stackrel{\rightharpoonup }{b}$两组向量$\stackrel{\rightharpoonup }{x_1},\stackrel{\rightharpoonup }{x_2},\stackrel{\rightharpoonup }{x_3},\stackrel{\rightharpoonup }{x_4},\stackrel{\rightharpoonup }{x_5}$和$\stackrel{\rightharpoonup }{y_1},\stackrel{\rightharpoonup }{y_2},\stackrel{\rightharpoonup }{y_3},\stackrel{\rightharpoonup }{y_4},\stackrel{\rightharpoonup }{y_5}$均由2个$\stackrel{\rightharpoonup }{a}$和3个$\stackrel{\rightharpoonup }{b}$排列而成.记$S=\stackrel{\rightharpoonup }{x_1}\stackrel{\rightharpoonup }{y_1}+\stackrel{\rightharpoonup }{x_2}\stackrel{\rightharpoonup }{y_2}+\stackrel{\rightharpoonup }{x_3}\stackrel{\rightharpoonup }{y_3}+\stackrel{\rightharpoonup }{x_4}\stackrel{\rightharpoonup }{y_4}+\stackrel{\rightharpoonup }{x_5}\stackrel{\rightharpoonup }{y_5}$，$S_{min}$表示$S$所有可能取值中的最小值.则下列命题的是（写出所有正确命题的编号）.
①$S$有5个不同的值.
②若$\stackrel{\rightharpoonup }{a}\perp \stackrel{\rightharpoonup }{b}$则$S_{min}$与$\left|\stackrel{\rightharpoonup }{a}\right|$无关.
③若$\stackrel{\rightharpoonup }{a}\parallel \stackrel{\rightharpoonup }{b}$则$S_{min}$与$\left|\stackrel{\rightharpoonup }{b}\right|$无关.
④若$\left|\stackrel{\rightharpoonup }{b}\right|>4\left|\stackrel{\rightharpoonup }{a}\right|$，则$S_{min}>0$.
⑤若$\left|\stackrel{\rightharpoonup }{b}\right|=2\left|\stackrel{\rightharpoonup }{a}\right|,S_{min}=8{\left|\stackrel{\rightharpoonup }{a}\right|}^2$，则$\stackrel{\rightharpoonup }{a}$与$\stackrel{\rightharpoonup }{b}$的夹角为$\frac{\pi }{4}$