Задача по SQL

CREATE TABLE IF NOT EXISTS "events" (
"id" INTEGER,
"cid" INTEGER NOT NULL,
"type" TEXT NOT NULL,
"date" DATETIME NOT NULL,
PRIMARY KEY("id")
);
INSERT INTO "events" ("id","cid","type","date") VALUES (1,0,'start','2021-01-01 00:00:00'),
(2,2,'start','2021-01-01 02:00:00'),
(3,1,'start','2021-01-01 03:00:00'),
(4,0,'end','2021-01-01 06:00:00'),
(5,1,'end','2021-01-01 07:00:00'),
(6,3,'start','2021-01-01 08:00:00'),
(7,3,'end','2021-01-01 08:30:00'),
(8,4,'start','2021-01-01 08:45:00'),
(9,2,'end','2021-01-01 09:00:00'),
(10,5,'start','2021-01-01 10:00:00'),
(11,6,'start','2021-01-01 11:00:00'),
(12,4,'end','2021-01-01 12:00:00'),
(13,5,'end','2021-01-01 13:00:00'),
(14,7,'start','2021-01-01 14:00:00'),
(15,7,'end','2021-01-01 15:00:00'),
(16,6,'end','2021-01-01 16:00:00'),
(17,8,'start','2021-01-01 15:30:00'),
(18,9,'start','2021-01-01 16:30:00'),
(19,10,'start','2021-01-01 17:00:00'),
(20,8,'end','2021-01-01 18:00:00'),
(21,10,'end','2021-01-01 19:00:00'),
(22,11,'start','2021-01-01 19:30:00'),
(23,11,'end','2021-01-01 19:45:00'),
(24,9,'end','2021-01-01 19:59:00');

Нужно составить запрос, который вернет начало и конец первого диапазона с максимальным количеством одновременных процессов.
Для этой таблицы это будет:
{start: "2021-01-01 03:00:00", end: "2021-01-01 06:00:00", count: 3}

 WITH 

    STARTENDS 

    AS 

        (SELECT E1."date" START_TIME, E2."date" END_TIME, E1."cid" CID 

           FROM "events" E1 INNER JOIN "events" E2 ON E1."cid" = E2."cid" 

          WHERE E1."type" = 'start' AND E2."type" = 'end'), 

 WITH events_cte AS ( 

    SELECT date, SUM(CASE WHEN type = 'start' THEN 1 ELSE -1 END) AS change 

    FROM events 

    GROUP BY date 

), 

cumulative_cte AS ( 

    SELECT date, SUM(change) OVER (ORDER BY date) AS cumulative 

    FROM events_cte 

), 

max_cte AS ( 

    SELECT MAX(cumulative) AS max_cumulative 

    FROM cumulative_cte 

), 

start_cte AS ( 

    SELECT MIN(date) AS start_date 

    FROM cumulative_cte 

    WHERE cumulative = (SELECT max_cumulative FROM max_cte) 

), 

end_cte AS ( 

    SELECT MIN(date) AS end_date 

    FROM cumulative_cte 

    WHERE date > (SELECT start_date FROM start_cte) 

) 

SELECT start_date AS start, end_date AS end, max_cumulative AS count 

FROM start_cte, end_cte, max_cte;