Can u help me II

Let $\text{Perm}(X)$ be the group of permutations on a set $X$ . Let $d\in\mathbb Z$ and $d\ge 2$ . The group $W=\{g\in\text{Perm}(\mathbb Z)~|~g(i+2d+2)=g(i)+2d+2,g(-i)=-g(i)\text{ for all }i\in\mathbb Z\}$ is called affine Weyl group. Any element $w\in W$ is uniquely determined by its value on $\{1,2,\dots,d\}$ , and we shall denote $w=[w(1),w(2),\dots,w(d)]$ . It's well known that $W$ is generated by $\{s_0,s_1,\dots,s_d\}$ , where $\begin{aligned} &s_0=[-1,2,\dots,d-1,d],\\ &s_d=[1,2,\dots,d-1,d+2],\\ &s_i=[1,\dots,i-1,i+1,i,i+2,\dots,d],\text{ for }i=1,\dots,d-1. \end{aligned}$ Is the following statement correct?

If $w=s_{i_1}s_{i_2}\dots s_{i_k}\in W$ with $s_{i_k}=s_0$ and $k$ as small as possible, then $w(1)<0$ .

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