Too Much Fractions

Algebra Level 4

\[ \left( \dfrac{55}{61} \right)^x + \left( \dfrac{56}{61} \right)^x + \cdots + \left( \dfrac{60}{61} \right)^x = \left( \dfrac{62}{61} \right)^x + \left( \dfrac{63}{61} \right)^x + \cdots + \left( \dfrac{65}{61} \right)^x + 1 \]

If \(x\) satisfies the equation above, find the value of \((x^2+1)^4 + (x+23)^2 + (x+1)^6 + 36\).

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