Russian Roulette — Re-Spinning

A 6-chamber revolver has its chambers arranged in a circle, numbered 1 through 6. A single bullet is placed into one of the 6 chambers. Two players take turns pulling the trigger — with the gun pointed at their own head. The barrel is spun again after every trigger pull, so each pull is independent. The person who gets killed loses.

Will you choose to be the first or the second player? What is the first player's probability of losing?