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Algebra Level 5

The smallest positive integral value of $a$ for which the greater root of the equation $x^2-(a^2+a+1)x+a(a^2+1)=0$ lies between the roots of the equation $x^2-a^2x-2(a^2-2)=0$ is:

\sqrt{2+\sqrt{2+\sqrt{2+\cdots}}}

\sqrt{\dfrac{27}{\sqrt{\dfrac{27}{\sqrt{\dfrac{27}{\sqrt{\cdots}}}}}}}

\sqrt 5\sqrt{\sqrt 5\sqrt{\sqrt 5\sqrt{\sqrt 5\sqrt{\cdots}}}}

\sqrt{4\sqrt{4\sqrt{4\sqrt{\cdots}}}}