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A $3$-input majority gate is defined by the logic function $M(a,b,c)=ab+bc+ca$. Which one of the following gates is represented by the function $M(\overline{M(a,b,c)}, M(a,b,\overline{c}),c)$?

1. $3$-input NAND gate
2. $3$-input XOR gate
3. $3$-input NOR gate
4. $3$-input XNOR gate