Infimum And Supremum In Triangles?

Geometry Level 5

Given a non-degenerate triangle \(ABC\) with perimeter 16 with \(A_1\), \(B_1\) and \(C_1\) on \(BC\), \(CA\) and \(AB\) such that \(AA_1\), \(BB_1\) and \(CC_1\) are medians.

What is the sum of the infimum and supremum values of \(AA_1+BB_1+CC_1\).

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