Aggregate overlapping date intervals

I have a table with some accounts, and their subscription's start and end dates. However, these subscriptions overlap sometimes and I need the start and end dates of every connected subscription period. Like in the example image.

I tried to merge the subscription periods with a date reference table and marked the dates if there was a subscription. However, the code was getting quite complex. I guess there must be a simpler solution.

IF OBJECT_ID('tempdb..#Subscriptions') IS NOT NULL DROP TABLE #Subscriptions
CREATE TABLE #Subscriptions (
    account_id varchar(1)
    ,start_date date
    ,end_date date
)

INSERT INTO #Subscriptions (account_id, start_date, end_date) values
('A','2019-06-20','2019-06-29'),
('A','2019-06-25','2019-07-25'),
('A','2019-07-20','2019-08-26'),
('A','2019-12-25','2020-01-25'),
('A','2021-04-27','2021-07-27'),
('A','2021-06-25','2021-07-14'),
('A','2021-07-10','2021-08-14'),
('A','2021-09-10','2021-11-12'),
('B','2019-07-13','2020-07-14'),
('B','2019-06-25','2019-08-26')

Directly:
WITH
cte1 AS (
SELECT account_id, start_date the_date, 1 weight
FROM Subscriptions
UNION ALL
SELECT account_id, end_date, -1
FROM Subscriptions
),
cte2 AS (
SELECT account_id,
the_date,
SUM(weight) OVER (PARTITION BY account_id
ORDER BY the_date, weight DESC) weight
FROM cte1
),
cte3 AS (
SELECT account_id,
the_date,
SUM(CASE WHEN weight = 0
THEN 1
ELSE 0
END) OVER (PARTITION BY account_id
ORDER BY the_date DESC) group_no
FROM cte2
)
SELECT account_id,
MIN(the_date) start_date,
MAX(the_date) end_date
FROM cte3
GROUP BY account_id, group_no
ORDER BY 1,2


https://dbfiddle.uk/?rdbms=sqlserver_2017&fiddle=82f72d57e6c527c3ce59d166c675e48c

StackExchange Database Administrators Q#316064, answer score: 8