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Algebra Level 4

Find the number of real values $x$ that satisfy the equation

$\lfloor x \rfloor + |x| + \lceil x \rceil =1000.$

Notations:

$| \cdot |$ denotes the absolute value function.
$\lfloor \cdot \rfloor$ denotes the floor function.
$\lceil \cdot \rceil$ denotes the ceiling function.

Special thanks to Jason Chrysoprase for inspiring this question.

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