JBMO-SL 2010 A2

Determine all four digit numbers $\overline{a b c d}$ such that

$$a(a+b+c+d)\left(a^{2}+b^{2}+c^{2}+d^{2}\right)\left(a^{6}+2 b^{6}+3 c^{6}+4 d^{6}\right)=\overline{a b c d}$$