2016? Nah I Choose 4016!

a+1b+1c=40162007,1c+1b+1a=pqa+\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,pa,b,c,p and qq satisfy the system of equations above. If pp and qq are coprime, submit your answer as p+qp+q.

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