Chum Says "The next piece of ice the Runner encounters during this run has +2 strength. Do 3 net damage unless the Runner breaks all subroutines on that piece of ice."
Femme Fatale says "When you install Femme Fatale, choose an installed piece of ice. When you encounter that ice, you may spend 1 [Credits] per subroutine on that ice to bypass it."
And the FAQ says
When an effect allows the Runner to “bypass” a piece of ice, he immediately passes that ice and continues the run. Any subroutines on the ice bypassed are not broken. Bypassing prevents any unresolved conditional abilities that met their trigger condition when the ice was encountered from resolving.
So having typed all that all that, I am now confused, if you don't break chum, then the next ICE is archer which you have Femme Fataled. The FF card says you do encounter it, but then bypass it. Bypass says the subroutines are not broken.
So.. Do you still take the damage from chum if you do bypass this? Is the chum effect still active for the next ICE you encounter in the run? Is the chum effect now wasted because he bypassed the next ICE he encountered?