Let \(\left(B_{t}ight)_{t \in[0,1]}\) and \(\left(\beta_{t}ight)_{t \in[0,1]}\) be independent one-dimensional Brownian motions. Show that the following process is again a Brownian motion:

\[W_{t}:= \begin{cases}B_{t}, & \text { if } t \in[0,1] \\ B_{1}+t \beta_{1 / t}-\beta_{1}, & \text { if } t \in(1, \infty)\end{cases}\]