
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
To successfully pass the "Civilized City" acceptance inspection, our city plans to comprehensively update and renovate some of the drainage main channels in the urban area. In response to the needs of urban construction, the project needs to be completed within one month. There are two construction teams, Team A and Team B, interested in contracting this project. After investigation, it is known that Team B can complete this project alone in $1.5$ times the time it takes for Team A to complete it alone. If Teams A and B cooperate, they can complete the project in $12$ days.  
$(1)$ How many days does each of Team A and Team B need to complete this project alone?  
$(2)$ If Team A's daily construction cost is $40,000 yuan and Team B's daily construction cost is $30,000 yuan, please design a plan that can be completed on time and minimize the construction cost.

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