# Can you guess it?

Algebra Level 5

If the three real roots of the equation $7x^3+7x^2-7x+1=0$ are:

\begin{aligned} x_1 &=&\dfrac{a}{a-\sec\left(\dfrac{m \pi}{b}\right)} \\ x_2 &=&\dfrac{a}{a+\sec\left(\dfrac{n \pi}{b}\right)} \\ x_3&=&\dfrac{a}{a+\sec\left(\dfrac{p \pi}{b}\right)} \\ \end{aligned}

Where $x_1 < x_2 < x_3$; $a, b, m, n, p$ are positive integers with $\text{gcd}(m,b)=\text{gcd}(n,b)=\text{gcd}(p,b)=1$ and $2m,2n,2p \leq b$

Find $2b-(a+m+n+p)$.

Inspired by this and this.

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