Problem for MySQL connoisseurs

Good afternoon, here I am faced with a task that is simple in appearance, but for the solution of which you need to keep your brains going :)

So: we have the following table:

id	rating	place
one	86	Null
2	five	Null
3	five	Null
four	41	Null
five	one	Null
6	91	Null

It is necessary to place a place (place) in the rating of each record, based on points (rating), and if the points are the same, use the secondary sign (id in this case). As a result, you need to get the following table:
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id	rating	place
one	86	five
2	five	2
3	five	3
four	41	four
five	one	one
6	91	6

Important note: the answer should be one UPDATE query (you can use nested SELECT queries).

Waiting for suggestions :)