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14-50 Plug Wire Run

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I don't see a problem with this, as long as there's enough room and you use the right type of wire. It may be possible for you to use the existing wire to thread and replace the original one (just in case) plus add the new 14-50 wire too. You should also check with a good electrician. They often allow free estimates.
 
You can, but you'll have to check some things. The NEC requires that when more than 3 current-carrying conductors are located in the same conduit, that you de-rate the conductors. 4-6 conductors require de-rating to 80%.

This will be okay for your #6, because even at 80% it's good to 60A. But the real question will be whether your conductors for your air conditioner will still have enough capacity when de-rated. It's likely they're not.

Be sure that you use listed electrical fittings for tees.