
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Recently, in order to jointly carry out the work of creating a national civilized urban area in Jiangjin District and consolidating the national health area, the Jiangjin District Construction Committee plans to green a $1200m^{2}$ area in a certain community. The construction will be completed by two construction teams, Team A and Team B, through bidding. It is known that Team A can complete a greening area of $100m^{2}$ per day, while Team B can complete a greening area of $50m^{2}$ per day. Let $x$ be the number of days Team A works first, and $y$ be the number of days Team B works next to exactly complete the greening task.<br/>$(1)$ Find the functional relationship of $y$ with respect to $x$;<br/>$(2)$ If Team A needs to pay a daily wage of $4000$ yuan to workers, and Team B needs to pay a daily wage of $3000$ yuan to workers, and if the number of days Team A works does not exceed the number of days Team B works, and is not less than $3$ days, how should Team A and Team B arrange their work days to minimize the total wages paid to the workers? Also, find the minimum total wage paid.

Present the answer in LaTex format: \boxed{Your answer}
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