How to get max value of a column when ids are unique but they are related through different variables

Question

with a1 as (
select 1 as user_id,'a' as v1,'x' as v2,1 as target_variable
union all
select 2,'a','y',2
union all
select 3,'b','y',3
union all
select 4,'b','z',4
union all
select 5,'c','z',5
union all
select 6,'d','x',6
union all
select 7,'f','w',7
),

a2 as (
select user_id,v1 as v1_v2_union,target_variable
from a1
union all
select user_id,v2,target_variable
from a1
),

a3 as (
select v1,string_agg(v2) as v2s,concat(v1,',',string_agg(v2)) as v1_v2
from a1
group by 1
),

a5 as (
select v1,v1_v2,max(target_variable) as target_variable
from(
select *
from a3
left join a2 on a3.v1_v2 like concat('%',a2.v1_v2_union,'%'))
group by 1,2
)

select user_id,max(target_variable) as target_variable
from(
select user_id,a5.*
from a5
left join a1 on v1_v2 like concat('%',a1.v1,'%') or v1_v2 like concat('%',a1.v2,'%')
)
group by user_id
order by user_id

If you take the above data as input you can see that user_id from 1 to 6 are related through v1 and v2 I am trying to find the max(target_variable) considering the relation between the ids

After running the above code I am getting the following as my output

Output

user_id	target_variable
1	6
2	6
3	6
4	5
5	5
6	6
7	7

The Output should be for user_1d 1 to 6 the max(target_variable)=6 and for user_id=7 the max(target_variable)=7

user_id	target_variable
1	6
2	6
3	6
4	6
5	6
6	6
7	7