Given a triangle \(ABC\) with circumcenter \(O\), how many moves does it take to draw a line \(\ell\) through \(O\) where \(\ell\) intersects \(AB\) and \(AC\) at \(P\), \(Q\) such that \(PO=OQ\)?

All terminology in this question is explained in the first note of my straightedge and compass set. More straightedge and compass constructions can be found there.

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