Two Date Range Lookup based on Index Formula

I'm currently working on this date range formula to capture a project name based on the date range. The formula works but only to a certain extent; I have 3 projects in July of 2021 and this formula gives me the 3 projects and then additional projects from other months.

I should only have 3 project totals, not a whole list of projects or at times only one project for July. The end date is not a concern as the formula is only to capture Column W as the start date for the project.

Project List:
Column H Column W Column Y
M2 7/21/2021 9/10/2021
M4 7/23/2021 9/10/2021
WC1 7/28/2021 9/1/2021

Here is my formula:

=IFERROR(INDEX('Project Name 21-22'!$H$3:$H$313,AGGREGATE(15,6,ROW('Project Name 21-22'!$W$3:$W$313)/(('Project Name 21-22'!$W$3:$W$313>=$F$3)+('Project Name 21-22'!$W$3:$W$313<=$H$3)*('Project Name 21-22'!$W$3:$W$313=$Z$5)+('Project Name 21-22'!$W$3:$W$313<=$H$3)*('Project Name 21-22'!$W$3:$W$313>"")>0),ROWS(Z$5:Z5))),"")

Dates are located in Cell F3 for 6/1/2021, H3 for 7/1/2021, and J3 for 8/1/2021

I can't figure out why this formula won't display only the 3 projects for July 2021 based on July 1 beginning date and Aug 1 as the end date.

Thanks!!!!

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