dollar-quoting in psql and in general

While trying to understand dollar-quoting, I found the following in psql:

select $a$ hello $a$;

behaves as expected, but psql does not like

select $a$ \ $a$;
or
select $a$ \\ $a$;

Should it? How should a dollar-quote handle:

$a$ \$a\$a $a$

?

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