Is it possible to reassign a mysql user variable different last_insert_ids?

Table a has a one-to-many relationship with table b.

table a (id)
table b (id, aId -> a.id)

Both table a and table b have autoincrementing ids.

I run the following statements in a single call:

insert into a values ();
set @id = last_insert_id();
insert into b (aId) values (@id);

insert into a values ();
set @id = last_insert_id();
insert into b (aId) values (@id);

insert into a values ();
set @id = last_insert_id();
insert into b (aId) values (@id);

...and here is the result:

select * from b
id  aId
1   3
2   3
3   3

This is not what I was expecting. Here's what I would expect to see:

id  aId
1   1
2   2
3   3

So it seems like only the last last_insert_id is used? After some experimenting, I find I can get the results I want, if I use a different variable name after each insert.

insert into a values ();
set @id1 = last_insert_id();
insert into b (aId) values (@id1);

insert into a values ();
set @id2 = last_insert_id();
insert into b (aId) values (@id2);

insert into a values ();
set @id3 = last_insert_id();
insert into b (aId) values (@id3);

However I was hoping to just have a single variable name, and sequentially reassign it as needed.

Is this possible?