If you receive a customer complaint, what is the first action you do?

Check the validity of the complaint...whether it is a valid complaint or not..

first I would calm the customer down and give him/her the feeling that the complaint was heard, understood and will be processed, meanwhile, I would see if the customer was telling the truth, and then if true i would satisfy the customer and provide an extra service as an apologize, if the complaint wasn't true, I would just explain politely the way of the service and remove any misunderstanding

First acknowledge the complaint and immediate response to customer & explain our interim action plans to avoid their line stoppage.

Check the background for the actual case.
Before you answer you need to go through the history to see what led to the complaint.
Giving the customer a holding note to assure them that we are working on the resolution maybe tempting, but in many cases it leads to impatience so before you do anything you need to be clear on the background and ready for any further questions and in depth analysis.

Go through the complaint first.