amount of days between a start date and end date within a particular quarter

A few quick admissions... I'm not an expert in Excel but I use it daily. I've tried with multiple AIs but either they don't get it or I'm not explaining it very well :-(

The issue..

I have a start date in K1 and an end date in L1.
Please reference the quarters (column A) and their years (Column B) as follows:
A6 = 1, B6 = current year,
A7 = 2, B7 = current year,
A8 = 3, B8 = current year,
A9 = 4, B9 = current year,
A10 = 1, B10 = current year +1,
A11 = 2, B11 = current year +1,
A12 = 3, B12 = current year +1,
A13 = 4, B13 = current year +1.

I want to be able to see the amount of days from the start date(K1) and end date(L1) that fall within a particular quarter.

In Excel, how to work out how many days a specific date range has in each quarter. The date range might span from one year to the next i.e. 1 Sept 2023 > 1 Feb 2024.

Any help and/or guidance would be SO appreciated!