"The Limited Liability Company Law and Partnership Law include provisions regarding the publication required upon the formation or authorization of domestic and foreign limited liability companies, limited partnerships and limited liability partnerships (hereinafter referred to as 'limited liability entities'). The applicable statutory provisions should be consulted prior to publishing.
Limited liability entities that are formed or authorized to do business in New York after June 1, 2006, which fail to comply with the publication requirements within 120 days after their formation or qualification will have their authority to carry on, conduct or transact any business suspended. "
TL; DR Do not violate the publication requirement. When suing a New York LLC, a common tactic is to show this requirement was not met. That then retroactively dissolves the LLC and lets you pierce through to the owner's assets.
Disclaimer: I am not a lawyer. Do not take legal advice from my Internet comments.