Set NULL value to a Customer Info field

I'd like to be able to write a NULL value to a Customer info field. Here's the situation:

I created a few lookup fields in Customer Info that get populated by a rule to mirror the assigned service team members in the Account Teams object. I had to flatten into Customer Info to be able to use these fields in Gainsight - for assigning a CTA, for example. In case a user gets removed from the account team, I'll need to blank out the field in Customer Info, but I can't figure out how to set a Null value with a rule. The "Set Action" page doesn't let me assign a blank value, and doesn't seem to work when I enter "NULL". 

Does anyone know any tricks to this? My plan is to have two rules - one to wipe out all the values of these fields, and then a second rule to populate them, which would be scheduled to run in that order. 

Thanks!