Sorting an array without using the excel Data menu, Filter, and Advanced. Formulas only, please.

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I have a question about sorting a list and removing blank cells vertically. what I have is a basic spreadsheet, it is built into my software so it does most all of excel's formulas, however, I am unaware that it support VBA code nor does it allow sorting and removing blanks like using Data tab, Filter Menu, and using the Advanced excel menu to do the sort. I was able to do this in excel with menu items. here is the original list.

I would like to transpose it to removing the blanks like this

but with using formulas in excel and not menu items.
Thank you to anyone who can help.