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I need to join my fence with my neighbor's; Can't use his post..how do I dig the hole for mine close to his?

As shown in the picture my neighbor's fence is already in place (the vertical fence) Mine will be horizontal and I need to set the post as close as possible to his. My post will support a gate (72" long)

His post (white) is already rotten and it is moving. What options do I have? How do I dig a hole next to his post (set in concrete) enter image description here

I don't think that his post will be enough to support everything. One option will be to extract his post, to replace it and to add a new one for myself and bury them both together in a large hole. Would this work?