Extracting Row and Column Header Coordinates from a value search

Hello Community,

I hope I can explain the situation well, and hopefully someone is able to help. If the could be longer I would have included "when multiple cells can contain the same value"

I have a very large dataset that I would like to apply the following searches to, but I will include a smaller sample that gets at the point.

For example; in the following sheet:
Cat Dog Lizard
Red 1 2 3
Yellow 4 5 6
Green 7 8 1

If I search for "2", I would like to get back "Red" and "Dog". I was able (by looking at other posts) come up with a series of formula that allow me to obtain the sheet coordinates and convert them into column and row headers (using SUMPRODUCT Row/Column); however, when multiple cells contain the same value, the output becomes the sum of that coordinate and is no longer a reference to a specific header, so a search for "1" would not give me "Red" and "Cat". Ideally, if I searched for "1" I would get back "Red" & "Cat" as well as "Green" & "Lizard", but I would be satisfied with a formula that even just gave me back one of the two header pairs.

Is what I'm trying even possible?

Much thanks in advance,

JordanSample Workbook.xlsx

If you insist with formula, please see the file

Regards
Azumi

Thank you all for your very timely responses. Each response was helpful both in solving the problem and furthering my understanding of excel.