Can you guess it?

Algebra Level 5

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

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

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 \(2b-(a+m+n+p)\).

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