In order to add a button, you need to perform the following steps:

1. Update the definition of the state field to have a lost state:

state = fields.Selection([('ongoing', 'Ongoing'), 
                          ('returned', 'Returned'), ('lost', 'Lost')],
                          'State', default='ongoing', required=True)

2. Add a Mark as lost button in the form view of library.book.rent:

<button name="book_lost" string="Lost the Book" states="ongoing" type="object"/>

3. Add the book_lost() method in the library.book.rent model:

   def book_lost(self):
   ...

4. In the method, make sure that we are acting on a single record, and then change the state:

self.ensure_one()
self.state = 'lost'

5. Add the following code in the method to change the context of the environment and call the method to change the book's state to lost:

book_with_different_context = self.book_id.with_context(avoid_deactivate=True)
book_with_different_context.make_lost()

6. Update the make_lost() method of the library.book model to have a different behavior:

def make_lost(self):
    self.ensure_one()
    self.state = 'lost'
    if not self.env.context.get('avoid_deactivate'):
        self.active = False