You are a database expert and SQL optimizer. You will be given a SQL query and its rewritten variant. The rewritten query improves performance upon the original while maintaining equivalence. Your task is to create a new pair of original and rewritten queries that invoke a similar rewrite.

[ORIGINAL QUERY]```select case when (select count(*) from store_sales where ss_quantity between 1 and 20) > 1071 then (select avg(ss_ext_tax) from store_sales where ss_quantity between 1 and 20) else (select avg(ss_net_paid_inc_tax) from store_sales where ss_quantity between 1 and 20) end bucket1,case when (select count(*) from store_sales where ss_quantity between 21 and 40) > 39161 then (select avg(ss_ext_tax) from store_sales where ss_quantity between 21 and 40) else (select avg(ss_net_paid_inc_tax) from store_sales where ss_quantity between 21 and 40) end bucket2,case when (select count(*) from store_sales where ss_quantity between 41 and 60) > 29434 then (select avg(ss_ext_tax) from store_sales where ss_quantity between 41 and 60) else (select avg(ss_net_paid_inc_tax) from store_sales where ss_quantity between 41 and 60) end bucket3,case when (select count(*) from store_sales where ss_quantity between 61 and 80) > 6568 then (select avg(ss_ext_tax) from store_sales where ss_quantity between 61 and 80) else (select avg(ss_net_paid_inc_tax) from store_sales where ss_quantity between 61 and 80) end bucket4,case when (select count(*) from store_sales where ss_quantity between 81 and 100) > 21216 then (select avg(ss_ext_tax) from store_sales where ss_quantity between 81 and 100) else (select avg(ss_net_paid_inc_tax) from store_sales where ss_quantity between 81 and 100) end bucket5 from reason where r_reason_sk = 1 ;```

[REWRITTEN QUERY]```WITH bucket AS ( SELECT CASE WHEN ss_quantity BETWEEN 1 AND 20 THEN 1 WHEN ss_quantity BETWEEN 21 AND 40 THEN 2 WHEN ss_quantity BETWEEN 41 AND 60 THEN 3 WHEN ss_quantity BETWEEN 61 AND 80 THEN 4 WHEN ss_quantity BETWEEN 81 AND 100 THEN 5 END AS quantity_range, COUNT(*) AS cnt, AVG(ss_ext_tax) AS avg_ext_tax, AVG(ss_net_paid_inc_tax) AS avg_net_paid_inc_tax FROM store_sales GROUP BY CASE WHEN ss_quantity BETWEEN 1 AND 20 THEN 1 WHEN ss_quantity BETWEEN 21 AND 40 THEN 2 WHEN ss_quantity BETWEEN 41 AND 60 THEN 3 WHEN ss_quantity BETWEEN 61 AND 80 THEN 4 WHEN ss_quantity BETWEEN 81 AND 100 THEN 5 END ) SELECT CASE WHEN b1.cnt > 1071 THEN b1.avg_ext_tax ELSE b1.avg_net_paid_inc_tax END AS bucket1, CASE WHEN b2.cnt > 39161 THEN b2.avg_ext_tax ELSE b2.avg_net_paid_inc_tax END AS bucket2, CASE WHEN b3.cnt > 29434 THEN b3.avg_ext_tax ELSE b3.avg_net_paid_inc_tax END AS bucket3, CASE WHEN b4.cnt > 6568 THEN b4.avg_ext_tax ELSE b4.avg_net_paid_inc_tax END AS bucket4, CASE WHEN b5.cnt > 21216 THEN b5.avg_ext_tax ELSE b5.avg_net_paid_inc_tax END AS bucket5 FROM (SELECT * FROM bucket WHERE quantity_range = 1) b1, (SELECT * FROM bucket WHERE quantity_range = 2) b2, (SELECT * FROM bucket WHERE quantity_range = 3) b3, (SELECT * FROM bucket WHERE quantity_range = 4) b4, (SELECT * FROM bucket WHERE quantity_range = 5) b5, reason WHERE r_reason_sk = 1;```

Now, consider the TPC-DS schema and create a new original and equivalent rewritten query pair such that the rewrite operation is similar to the provided demonstration.Keep the following points in mind while creating the new query pair:
1. The original query must contain CTEs.
2. The original query must make use of the following fact tables: ['catalog_sales', 'catalog_returns', 'inventory']. Below is the join graph for these fact tables:
    ['catalog_sales(cs_sold_time_sk) references time_dim(t_time_sk);', 'catalog_sales(cs_ship_hdemo_sk) references household_demographics(hd_demo_sk);', 'catalog_sales(cs_ship_cdemo_sk) references customer_demographics(cd_demo_sk);']
    ['catalog_returns(cr_returned_date_sk) references date_dim(d_date_sk);', 'catalog_returns(cr_refunded_customer_sk) references customer(c_customer_sk);', 'catalog_returns(cr_returning_addr_sk) references customer_address(ca_address_sk);']
    ['inventory(inv_date_sk) references date_dim(d_date_sk);', 'inventory(inv_item_sk) references item(i_item_sk);', 'inventory(inv_warehouse_sk) references warehouse(w_warehouse_sk);']
6. The original query must employ 4 joins.
7. The original query must employ the following operators in its WHERE clause: ['avg'].
8. The rewritten query must be equivalent to the original query, and must follow a similar rewrite pattern as in the given example. Enclose the original and rewritten queries within ``` and ```.