Complex Sum to Zero

Algebra Level 5

Find the number of different ordered quadruples (a,b,c,d)(a,b,c,d) of complex numbers such that {a2=1b3=1c4=1d6=1a+b+c+d=0\begin{cases} a^2=1\\ b^3=1\\ c^4=1\\ d^6=1\\ a+b+c+d=0 \end{cases}

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