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

[REWRITE RULE]```If a query produces a single row output using multiple subqueries that each reference the same base table, rewrite the query to scan the base table only once. Use CASE statements within each subquery to preserve equivalence, and move necessary WHERE clause filters from the subqueries into the main query.```

[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]```select case when count(case when ss_quantity between 1 and 20 then 1 end) > 1071 then avg(case when ss_quantity between 1 and 20 then ss_ext_tax end) else avg(case when ss_quantity between 1 and 20 then ss_net_paid_inc_tax end) end as bucket1, case when count(case when ss_quantity between 21 and 40 then 1 end) > 39161 then avg(case when ss_quantity between 21 and 40 then ss_ext_tax end) else avg(case when ss_quantity between 21 and 40 then ss_net_paid_inc_tax end) end as bucket2, case when count(case when ss_quantity between 41 and 60 then 1 end) > 29434 then avg(case when ss_quantity between 41 and 60 then ss_ext_tax end) else avg(case when ss_quantity between 41 and 60 then ss_net_paid_inc_tax end) end as bucket3, case when count(case when ss_quantity between 61 and 80 then 1 end) > 6568 then avg(case when ss_quantity between 61 and 80 then ss_ext_tax end) else avg(case when ss_quantity between 61 and 80 then ss_net_paid_inc_tax end) end as bucket4, case when count(case when ss_quantity between 81 and 100 then 1 end) > 21216 then avg(case when ss_quantity between 81 and 100 then ss_ext_tax end) else avg(case when ss_quantity between 81 and 100 then ss_net_paid_inc_tax end) end as bucket5 from store_sales where exists (select 1 from 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 adheres to the provided SQL rewrite rule.Keep the following points in mind while creating the new query pair:
1. The original query must make use of the following fact tables: ['catalog_sales', 'inventory']. Below is the join graph for these fact tables:
    ['catalog_sales(cs_ship_date_sk) references date_dim(d_date_sk);', 'catalog_sales(cs_ship_mode_sk) references ship_mode(sm_ship_mode_sk);', 'catalog_sales(cs_sold_date_sk) references date_dim(d_date_sk);']
    ['inventory(inv_date_sk) references date_dim(d_date_sk);', 'inventory(inv_warehouse_sk) references warehouse(w_warehouse_sk);', 'inventory(inv_item_sk) references item(i_item_sk);']
4. The original query must employ 1 joins.
5. The original query must employ the following aggregate functions: ['avg', 'max', 'sum'].
6. The original query must employ the following operators in its WHERE clause: [].
7. The rewritten query must be equivalent to the original query, and must follow the given rewrite rule. Enclose the original and rewritten queries within ``` and ```.