Let security summon the nanomed

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security officers are typically first to the scene of a murder or assault, so why not let them summon the nanomed so they can heal anyone hurt in the event?

Medkits were removed from security for exactly this reason, Security gets on scene first, or is the person injured, and thus any injures are now no longer a problem.
Medical heals people, Security can call, or bring people to medical, its not your job to patch up a person. Hard no.

(PS, there is almost nothing stopping you from either A. Getting the AI to send you the Nanomed program onto your PDA at the start of the shfit or B. Grabbing a MedU.PAK to put into your PDA and copy the file to you anyways. Go ahead, be a power nerd on Security, we love that.)