Suppose a line 'l' cuts an equilateral triangle \(ABC\) of side \( \sqrt{3}\) in two points different from the vertices. Say it cuts \(AB\) and \(AC\) in points \(R\) and \(Q\) respectively. We mark the orthocentre \(H\) of the triangle \(ARQ\) and also the midpoint \(M\) of side \(RQ\). We extend \(HM\) to a point \(T\) so that \(HM = MT\). The point \(P\) is the foot of perpendicular from \(T\) on side \(BC\).

Now, we draw outward equilateral triangles \(RA'Q\), \(QC'P\), and \(PB'R\). (The Napoleonic triangles of triangle \(PQR\)).

The task is to find: \(A'T + B'T + C'T\).

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