2016? Nah I Choose 4016!

Number Theory Level 2

\[a+\dfrac{1}{b+\dfrac{1}{c}} = \frac{4016}{2007} , \qquad \dfrac{1}{c+\dfrac{1}{b+\dfrac{1}{a}}} = \dfrac{p}{q}\]

Positive integers \(a,b,c,p\) and \(q\) satisfy the system of equations above. If \(p\) and \(q\) are coprime, submit your answer as \(p+q\).