Let \(P(x) =x^2-\left(\dfrac{a^2}{2}+5a+1\right)x+a+3\) be a quadratic polynomial such that \(P(x_1)=P(x_2)=0\) and \(x_1+x_2=13\).

Is there a real value for \(a\) for which \(x_1\) and \(x_2\) are natural numbers?

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