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ALBANY — New York lawmakers on Tuesday approved the most comprehensive set of gun bills in the state in six years, including measures that would ban bump stocks, prohibit teachers from carrying guns in schools and extend the waiting period for gun buyers who do not pass an instant background check.

In total, six gun bills passed easily through the State Senate and Assembly, a remarkable sight in a Capitol that for years had resisted almost all new legislation on the subject.

Gov. Andrew M. Cuomo, a Democrat, last ushered a major gun safety package into law in 2013, after the massacre at Sandy Hook Elementary School in Newtown, Conn. The governor successfully corralled recalcitrant Senate Republicans into supporting the so-called Safe Act that expanded the state’s ban on assault weapons, tightened certification requirements, increased criminal penalties for illegal guns and closed private sale loopholes.

Mr. Cuomo has described the Safe Act as one of his signature achievements. But it has also become a rallying cry for some gun owners and Republican politicians who felt it overreached.