The product of the roots of the quadratic equation \(2x^2 - 7x + 3 = 0\) is equal to the sum of another quadratic equation's roots, \(ax^2 + bx + c = 0\). If the roots of the second equation are also the roots of the polynomial \(y^2 - 5y + 6 = 0\), find the value of \(a + b + c\). Let's think step by step and output the final answer within \boxed{}.
user