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Let $r_1,r_2$ be the two largest real roots of the polynomial $P(x)=3x^3-17x+5\sqrt{6}$

If $r_1+ r_2$ can be expressed as $\dfrac{\sqrt{a}+\sqrt{b}}{c}$ for positive integers $a,b,c$, then what is the smallest possible value of $a+b+c$?

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