\(ABC\) is an equilateral triangle with side length \(6\)

Point \(E\) is on the external angle bisector from \(B\) such that \(BE=9\).

The perpendicular dropped from \(E\) to \(BC\) intersects \(BC\) at \(G\) and \(AC\) at \(F\).

Point \(I\) is on line \(BF\) extended through \(F\) such that \(\angle FIG=30^{\circ}\).

The circumcenter of \(\triangle BIG\) is \(O\)

Find \(8 \times AO\).

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