Organic groups menu synchronization

Synchronize Organic Groups with the menu: if a node is attached to the menu, all the menu parent links are checked. If these links lead to a group the node can be put into, the node is automagically put into the group. It works in the same way on menu link updates and deletions, to remove nodes from groups.
The usual OG audience fields are hidden, so that the only way to put a node in a group or to remove it from a group is to use the node menu UI: create, update or delete the node menu link so that parent menu links contain the group or not.
The menu administration UX is altered to hide menu links to non-accessible groups and their children.

Why this? For some websites, it can be a big UX improvement to handle easily in the same menu tree some public and private nodes.