Does the floor change anything?

Given $\large \sqrt{\left \lfloor 2 + \sqrt{\left \lfloor 2+ \sqrt{ \left \lfloor 2 + \sqrt{\left \lfloor 2+ ... \right \rfloor }\right \rfloor }\right \rfloor}\right \rfloor } = \zeta$

find $\left \lfloor 1000 \zeta \right \rfloor$ .