Alternating Exponential Prime Finite Sum Polynomial (n=7)

Number Theory Level pending

Let \(s_7(x)=7x^7+a_6x^6+a_5x^5-7^5x^4+7^4x^3-7^3x^2+7^2x-7\). Find \(a_6\) and \(a_5\) such that \(x=-\frac{1}{7}\) is a rational root and the sum of the coefficients is \(2(1+2+3+4+5+6+7)\).

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