Integer triangle altitudes

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 $AH, HD, BH, HE, CH, HF$ .