Does the ceiling change anything?

Algebra Level 3

Given 2+2+2+2+...=ζ \large \sqrt{\left \lceil 2 + \sqrt{\left \lceil 2+ \sqrt{ \left \lceil 2 + \sqrt{\left \lceil 2+ ... \right \rceil }\right \rceil }\right \rceil}\right \rceil } = \zeta

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

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