Return a Month When Dates Cross over Calendar Month

Excel 2010:
I need a formula that returns a particular month but because the date ranges cross over the calendar month, I cannot use the =TEXT(A2,"mmm") formula. For example, I need the formula to return JAN in cell A2 if the date in cell A1 is between 1/1/2016 and 1/29/2016. It should return FEB if the date in cell A1 is between 1/30/2016 and 2/26/2016. Also, the raw data was pulled down from another source and the date format is showing as [$-10409]m/d/yyyy. Does that matter?

Is there a complete pattern for all months ?
Do you have those date ranges stored in a range of cells ?

For my purposes, the last day of the month is the last Friday of each month. The next month starts on the very next calendar day and ends on the last Friday and so on.

I just wrote down the ranges to make it easier to reference for formula creation:

JAN 1/1/2016 - 1/29/2016
FEB 1/30/2016 - 2/26/2016
MAR 2/27/2016 - 3/25/2016
APR 3/26/2016 - 4/29/2016
MAY 4/30/2016 - 5/27/2016
JUN 5/28/2016 - 6/24/2016
JUL 6/25/2016 - 7/29/2016
AUG 7/30/2016 - 8/26/2016
SEP 8/27/2016 - 9/30/2016

So wouldn't January begin with the day immediately following the last friday of December 2015 ?
i.e. Dec 26th 2015

?

I had the very same thought lol! But I inherited this data so I am working with what I have. Besides when I checked the raw dataset, there was no data with date less than 1/1/2016 but to maintain the pattern you can change it to read

JAN 12/26/2015 - 1/29/2016

Try

=TEXT(WORKDAY.INTL(A2-1,1,"1111011"),"mmm")

I think that worked!!! I am still doing a lil spot checking but Eureka! Thank you.

You're welcome.