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Does the passive device need to start in the claw of the robot at the beginning of the competition or can it be placed in the robot starting box and picked up later in the competition?

No, it does not need to start in the claw. It does need to fit in the 30 cm box with the rest of the robot. The passive device will be considered part of the robot and scored as such if it is left in a different Zone than the Robot at the end of the competition. See Rule 6.a.v.