
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
Two construction teams jointly participate in a road construction project. Team A completes one-third of the total project in 30 days. Then, team B joins, and both teams work together for 15 days to complete the entire project.  
$(1)$ Find the number of days team B alone needs to complete the entire project.  
$(2)$ If team A works for 4 days and team B works for 3 days, the labor cost is 42,000 yuan. If team A works for 5 days and team B works for 6 days, the labor cost is 75,000 yuan. Find the daily labor cost for team A and team B.  
$(3)$ Under the conditions in $(2)$, if the two teams do not work simultaneously and the total labor cost does not exceed 280,000 yuan, how many days will it take to complete the entire project as quickly as possible?

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