Does the floor change anything?

Algebra Level pending

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 \( \left \lfloor 1000 \zeta \right \rfloor \).

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