JEE 2016! (will vectors help?)

Geometry Level 4

The Pentagon \(ABCDE\) has the properties \(AB || CD , ED || BC , \dfrac{AB}{CD} = \dfrac12 , \dfrac{ED}{BC} = \dfrac13 \).

Given that the straight lines \(AD\) and \(CE\) intersect at point \(O\) to form the ratios \( \dfrac{OC}{EO} = \dfrac ab \) and \(\dfrac{OD}{AO} = \dfrac cd \), where \(a,b,c,d\) are positive integers with \(\gcd(a,b) = \gcd(c,d) = 1 \).

Find the value of \(a+b+c+d\).

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