Customers active per month - Loop BIgQuery

I want to group the active customers per month based on the following condition included in the case statement below, ie bins per month.

I managed to get my results through a long case statement and UNION ALL but I believe there must be a smarter way to make this work with the help of a script and a loop through a dates reference table

Manual iteration query that worked:

SELECT customer_id WHEN ((DATE (start_time) >= '2022-03-01'
                          AND DATE(end_time) <= '2022-03-31')
                         OR (DATE(start_time) < '2022-03-01'
                             AND DATE_TRUNC(end_time,MONTH) = '2022-03-01')
                         OR (DATE(end_time)>'2022-03-31'
                             AND DATE_TRUNC(start_time,MONTH) ='2022-03-01')) THEN '2022-03-01' ELSE 'n.a' END AS MONTH ,
FROM customers
UNION ALL
SELECT customer_id CASE
                       WHEN ((DATE (start_time) >= '2022-04-01'
                              AND DATE(end_time) <= '2022-04-30')
                             OR (DATE(start_time) < '2022-04-01'
                                 AND DATE_TRUNC(end_time,MONTH) = '2022-04-01')
                             OR (DATE(end_time)>'2022-04-30'
                                 AND DATE_TRUNC(start_time,MONTH) ='2022-04-01')) THEN '2022-04-01'
                       ELSE 'n.a'
                   END AS MONTH ,
FROM customers 
...........
SELECT customer_id CASE
                       WHEN ((DATE (start_time) >= '2022-12-01'
                              AND DATE(end_time) <= '2022-12-31')
                             OR (DATE(start_time) < '2022-12-01'
                                 AND DATE_TRUNC(end_time,MONTH) = '2022-12-01')
                             OR (DATE(end_time)>'2022-12-31'
                                 AND DATE_TRUNC(start_time,MONTH) ='2022-12-01')) THEN '2022-12-01'
                       ELSE 'n.a'
                   END AS MONTH ,
FROM customers 

TLDR I want to create a loop that iterates through the dates_reference table, a script that creates these individual tables for me and unions them all in a final table.

dates_reference table

customers table

This would be my approach in code, but I do not know how to write it in BQ:

  start_date= first_day (loop trough the dates reference date )
  end_date = last_day (loop trough the dates reference date )
  query =f"""

SELECT customer_id WHEN ((DATE (start_time) >= start_date
                          AND DATE(end_time) <= end_date)
                         OR (DATE(start_time) < start_date
                             AND DATE_TRUNC(end_time,MONTH) = start_date)
                         OR (DATE(end_time)>end_date
                             AND DATE_TRUNC(start_time,MONTH) =start_date)) THEN start_date ELSE 'n.a' END AS MONTH ,  """