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 $p+q$ .