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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