Write SQL to compare exclusive category engagement

You are given session-level data and must compare engagement between users who exclusively used the 'social' category versus those who exclusively used the 'game' category in a fixed window. Use the 28-day window from 2025-08-04 to 2025-09-01 inclusive. Define two mutually exclusive cohorts: 'social_only' users had ≥1 'social' session and 0 'game' sessions in the window; 'game_only' users had ≥1 'game' session and 0 'social' sessions. Exclude everyone else (including users with both categories or only 'other'). Compute, for each cohort: (1) users_count; (2) median_active_days (median over users of distinct calendar days with ≥1 session); (3) pct_users_active_10_plus_days (share of users with active_days ≥ 10); and (4) avg_minutes_per_active_day (for each user, total duration_minutes in the window divided by that user's active_days; then average across users). Return one row per cohort with columns: cohort, users_count, median_active_days, pct_users_active_10_plus_days, avg_minutes_per_active_day. Assume standard SQL (e.g., PostgreSQL). Handle data quality by ignoring sessions with NULL category or duration_minutes ≤ 0. Schema and small sample data:

Schema: users(user_id INT, signup_date DATE) sessions(session_id INT, user_id INT, session_start TIMESTAMP, session_end TIMESTAMP, category VARCHAR, duration_minutes INT)

Sample tables (ASCII): users +---------+-------------+ | user_id | signup_date | +---------+-------------+ | 1 | 2025-07-15 | | 2 | 2025-08-10 | | 3 | 2025-08-20 | | 4 | 2025-08-25 | +---------+-------------+

sessions +------------+---------+---------------------+---------------------+----------+-----------------+ | session_id | user_id | session_start | session_end | category | duration_minutes| +------------+---------+---------------------+---------------------+----------+-----------------+ | 101 | 1 | 2025-08-05 10:00:00 | 2025-08-05 10:30:00 | social | 30 | | 102 | 1 | 2025-08-06 10:00:00 | 2025-08-06 10:20:00 | social | 20 | | 103 | 2 | 2025-08-07 09:00:00 | 2025-08-07 10:00:00 | game | 60 | | 104 | 2 | 2025-08-08 09:00:00 | 2025-08-08 09:45:00 | game | 45 | | 105 | 3 | 2025-08-09 12:00:00 | 2025-08-09 12:10:00 | social | 10 | | 106 | 3 | 2025-08-13 12:00:00 | 2025-08-13 12:10:00 | game | 10 | | 107 | 4 | 2025-08-11 14:00:00 | 2025-08-11 14:30:00 | game | 30 | +------------+---------+---------------------+---------------------+----------+-----------------+

Write a single SQL query (CTEs allowed) that produces the required output for the given window, correctly excluding mixed-category users and handling the data quality constraints.