# Going to be beautiful

Algebra Level 5

The equation $${ (z+1) }^{ 7 }+{ z }^{ 7 }=0$$ has roots $${ z }_{ 1 },{ z }_{ 2 }\cdots{ z }_{ 7 }.$$ Let $$\sum _{ r=1 }^{ 7 }{ \text{Re}({ z }_{ r }) } =a$$ and $$\sum _{ r=1 }^{ 7 }{ \text{Im}({ z }_{ r }) } =b.$$ Then the quadratic equation whose roots are $$\left| 2a \right|$$ and $$\left| b+9 \right|$$ can be represented as $${ x }^{ 2 }+cx+d=0.$$ Find the value of $$(\left| c \right| +\left| d \right| +21).$$

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