Actions inside a dropdown

We need to allow a user to select a project as primary and quick action, he should also be able to add a new project, edit and delete projects (as secondary actions) this is all done in the faceted menu, since selection is the most common action, we decided that a dropdown would be best:

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Now we are trying to figure out where we should place edit and delete, one option is to place it inside the dropdown.

Project selection

The issue is, it doesn't feel right to place buttons inside a dropdown, it is both unexpected and might confuse a user that is used to clicking the entire button in a dropdown. so I tried to find similar behaviors in other products:

I found a similar example of such behavior in trello

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Do you know any other products that has such behavior? Do you find this behavior usable and useful or can you suggest a better way to perform these tasks?

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