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 ca_state,cd_gender,cd_marital_status,cd_dep_count,count(*) cnt1,avg(cd_dep_count),stddev_samp(cd_dep_count),sum(cd_dep_count),cd_dep_employed_count,count(*) cnt2,avg(cd_dep_employed_count),stddev_samp(cd_dep_employed_count),sum(cd_dep_employed_count),cd_dep_college_count,count(*) cnt3,avg(cd_dep_college_count),stddev_samp(cd_dep_college_count),sum(cd_dep_college_count) from customer c,customer_address ca,customer_demographics where c.c_current_addr_sk = ca.ca_address_sk and cd_demo_sk = c.c_current_cdemo_sk and exists (select * from store_sales,date_dim where c.c_customer_sk = ss_customer_sk and ss_sold_date_sk = d_date_sk and d_year = 1999 and d_qoy < 4) and (exists (select * from web_sales,date_dim where c.c_customer_sk = ws_bill_customer_sk and ws_sold_date_sk = d_date_sk and d_year = 1999 and d_qoy < 4) or exists (select * from catalog_sales,date_dim where c.c_customer_sk = cs_ship_customer_sk and cs_sold_date_sk = d_date_sk and d_year = 1999 and d_qoy < 4)) group by ca_state,cd_gender,cd_marital_status,cd_dep_count,cd_dep_employed_count,cd_dep_college_count order by ca_state,cd_gender,cd_marital_status,cd_dep_count,cd_dep_employed_count,cd_dep_college_count limit 100;```

[REWRITTEN QUERY]```select ca_state, cd_gender, cd_marital_status, cd_dep_count, count(*) cnt1, avg(cd_dep_count), stddev_samp(cd_dep_count), sum(cd_dep_count), cd_dep_employed_count, count(*) cnt2, avg(cd_dep_employed_count), stddev_samp(cd_dep_employed_count), sum(cd_dep_employed_count), cd_dep_college_count, count(*) cnt3, avg(cd_dep_college_count), stddev_samp(cd_dep_college_count), sum(cd_dep_college_count) from customer c join customer_address ca on c.c_current_addr_sk = ca.ca_address_sk join customer_demographics cd on cd.cd_demo_sk = c.c_current_cdemo_sk where c.c_customer_sk in ( select ss_customer_sk from store_sales join date_dim on ss_sold_date_sk = d_date_sk where d_year = 1999 and d_qoy < 4 ) and ( c.c_customer_sk in ( select ws_bill_customer_sk from web_sales join date_dim on ws_sold_date_sk = d_date_sk where d_year = 1999 and d_qoy < 4 ) or c.c_customer_sk in ( select cs_ship_customer_sk from catalog_sales join date_dim on cs_sold_date_sk = d_date_sk where d_year = 1999 and d_qoy < 4 ) ) group by ca_state, cd_gender, cd_marital_status, cd_dep_count, cd_dep_employed_count, cd_dep_college_count order by ca_state, cd_gender, cd_marital_status, cd_dep_count, cd_dep_employed_count, cd_dep_college_count limit 100;```

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: ['web_sales', 'inventory']. Below is the join graph for these fact tables:
    ['web_sales(ws_sold_date_sk) references date_dim(d_date_sk);', 'web_sales(ws_ship_cdemo_sk) references customer_demographics(cd_demo_sk);', 'web_sales(ws_ship_customer_sk) references customer(c_customer_sk);']
    ['inventory(inv_item_sk) references item(i_item_sk);', 'inventory(inv_warehouse_sk) references warehouse(w_warehouse_sk);', 'inventory(inv_date_sk) references date_dim(d_date_sk);']
5. The original query must employ 3 joins.
6. The original query must employ the following aggregate functions: ['avg'].
7. The original query must employ the following operators in its WHERE clause: ['avg', 'sum'].
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 ```.