You can enter the factories relatively low, like 40+ laffs, but going into Cog HQ itself without at least 7 very strong toons with a 62 laff might prove difficult. I would be at least 80+ if I were you. But the wonderful thing is that you can keep on trying no matter what your laff.
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My doctor gave me pills to help with my memory, but I keep forgetting to take them. 
WHO ARE YOU? WHY DO YOU KEEP BUGGIN' ME!?
WAIT...WHO AM I?