Define the floor for a real number, \( x \), to be the the largest integer \( \le x \). We write the floor of \( x \) as \( \lfloor x \rfloor \).
Define the ceiling for a real number, \( x \), to be the smallest integer \( \ge x \). We write the ceiling of \( x \) as \( \lceil x \rceil \).
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