Integer triangle altitudes

Geometry Level 3

In triangle \(ABC,\) the feet of the perpendiculars from \(A, B, C\) are \(D, E, F,\) respectively, and the following holds: \[AH \le BH \le CH,\] where\(H\) is the orthocenter of the triangle.

Find the specific triangle in which the segments \(AH, BH, CH, HD, HE, HF\) all have integer lengths, \(AH\) and \(HD\) have an integer geometric mean, and \(AD+BE+CF\) is as small as possible.

Then, enter the concatenated value of \(AH, HD, BH, HE, CH, HF\).

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