Does the floor change anything?

Algebra Level 4

Given 2+2+2+2+...=ζ \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 1000ζ \left \lfloor 1000 \zeta \right \rfloor .

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