Postgres: Detect active and inactive periods

I have a table containing events. Each event has a defined starting and ending timestamp:

From this structure, I want to find out, active and inactive periods. Active periods are defined by the time range in which an event takes place. Such an active period can be composed of multiple consecutive events.
In inactive periods no event is taking place. So the expected output would be like:

Any idea how I can achieve this by using PostgreSQL?

CREATE TABLE IF NOT EXISTS events(
   id INT PRIMARY KEY,   
   start_ts timestamp NOT NULL,
   end_ts timestamp NOT NULL
);

INSERT INTO events(id,start_ts,end_ts)
VALUES
(1,'2023-07-27 01:02:00','2023-07-27 01:05:00'),
(2,'2023-07-27 01:05:00','2023-07-27 01:07:00'),
(3,'2023-07-27 01:07:00','2023-07-27 01:11:00'),
(4,'2023-07-27 01:11:00','2023-07-27 01:15:00'),
(5,'2023-07-27 01:30:00','2023-07-27 01:35:00'),
(6,'2023-07-27 01:35:00','2023-07-27 01:42:00'),
(7,'2023-07-27 01:45:00','2023-07-27 01:50:00');

Another task about gaps and islands. Additionally, with adding gaps to output. Only for case The events cannot overlap each other
See this example.

with t1 as(-- gaps
  select *
  ,case when lag(end_ts,1,start_ts)over(order by start_ts)=start_ts then 0 else 1 end gap
from events
)
,t2 as(--count islands
  select *
    ,sum(gap)over(order by start_ts)gr
  from t1
)
,t3 as(-- compact islands and take time for gap
  select min(start_ts)start_ts,max(end_ts)end_ts,gr
    ,lead(min(start_ts))over(order by gr) next_active
  from t2
  group by gr
)
select start_ts,end_ts,gr,'active' as status
from t3
union all
select end_ts start_ts,next_active end_ts,gr,'inactive' as status
from t3
where next_active is not null
order by gr,start_ts

Result