Algebra Level 5

For constant $$a$$, let the roots (of $$x$$) of the equation $$x^2+ 2(a-3) x+9= 0$$ lie between $$-6$$ and $$1$$.

Given that

$$2, h_1, h_2, h_3,\ldots,h_{20} , \lfloor a\rfloor$$ follows a harmonic progression.
$$2, a_1, a_2, a_3,\ldots,a_{20} , \lfloor a\rfloor$$ follows an aritmetic progression.

Find the value of $$a_3 \times h_{18}$$.

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