It's All Squares

$\tan(60^{\circ}) = \sqrt{3} = \dfrac{a}{x} \implies x = \dfrac{a}{\sqrt{3}}$

and

$\tan(60^{\circ}) = \sqrt{3} = \dfrac{3a}{y} \implies y = \sqrt{3}a$

$\implies 1 = \dfrac{a}{\sqrt{3}} + 6a + \sqrt{3}a \implies 2(2 + 3\sqrt{3})a = \sqrt{3} \implies$

$a = \dfrac{\sqrt{3}}{2(2 + 3\sqrt{3})} = \dfrac{\sqrt{3}(3\sqrt{3} - 2)}{2(23)} = \boxed{\dfrac{9 - 2\sqrt{3}}{46}}$

$\implies A = 14a^2 = 14(\dfrac{9 - 2\sqrt{3})}{46})^2 = \dfrac{21(31 - 12\sqrt{3})}{1058} =$ $\dfrac{a^{*}(b - c\sqrt{d})}{e}$

$\implies a^{*} + b + c + d + e = \boxed{1125}$ .