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Let the point \( B \) be the reflection of the point \( A(2,3) \)
\( \mathrm{P} \) with respect to the line \( 8 x-6 y-23=0 \). Let \( \Gamma_{A} \) and

W \( \Gamma_{B} \) be circles of radii 2 and 1 with centres \( A \) and \( B \) respectively. Let \( T \) be a common tangent to the circles \( \Gamma_{A} \) and \( \Gamma_{B} \) such that both the circles are on the same side of \( T \). If \( C \) is the point of intersection of \( T \) and the line passing through \( A \) and \( B \), then the length of the line segment \( A C \) is
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