Sort using AGGREGATE

Is it possible to use the AGGREGATE function to sort data without using helper columns?

I have two columns of data (numbers and letters) from A1:B5

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In C1:D5 I use the AGGREGATE formula (this formula is my own variation) to list non blank values that are in column B

IFERROR(INDEX($A$1:$B$5,AGGREGATE(15,6,ROW($A$1:$B$5)/(($B$1:$B$5<>"")),ROWS(A$1:A1))-(1-1),COLUMN()-2),"")

So I get

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I was hoping I could do this

IFERROR(INDEX($A$1:$B$5,AGGREGATE(15,6,ROW($A$1:$B$5)/(($B$1:$B$5<>"")*($A$1:$A$5={1,2,3,4,5})),ROWS(A$1:A1))-(1-1),1),"")

dictating the order I want the output in to end up with, effectively sorting the output

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Is there any way to use the AGGREGATE formula to dictate what order you want the results in and doing it within one formula?

Try this
In F2

=IFERROR(AGGREGATE(15,6,$A$2:$A$6/($B$2:$B$6<>""),ROWS($F$2:$F2)),"")

In G2

=IFERROR(INDEX($B$2:$B$6,MATCH(F2,$A$2:$A$6,0)),"")

then drag down.

Thanks guys, was hitting a brick wall over this one!