Victor's Level 5 Challenges - 3

Algebra Level 5

There exists a constant \(C\) such that

\[\sum_{i=1}^4\left(x_i+\frac{1}{x_i}\right)^3 \geq C\]

for all positive real numbers \(x_1,x_2,x_3,x_4\) which satisfy

\[x_1^3+x_3^3+3x_1x_3=x_2+x_4=1.\]

Given that the largest constant \(C\) can be expressed in the form \(\frac{m}{n}\), where \(m\) and \(n\) are coprime, positive integers, find the value of \(m+n\).

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