
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
From August 27th to 29th, 2022, the World Clean Energy Equipment Conference with the theme of "New Energy, New Intelligent Manufacturing, New Era" was held in Deyang. The conference focused on the hotspots and forward-looking issues of the clean energy equipment industry, aiming to achieve industry convergence through exhibition gatherings. The Deyang Clean Energy Equipment characteristic town is located in the Deyang Economic and Technological Development Zone, with a planned area of 4.82 square kilometers and is scheduled to be basically completed by 2025. If two construction teams, Team A and Team B, plan to participate in the construction of a project in the "characteristic town," it is known that Team A alone needs 18 months to complete the task. If Team B works alone for 2 months first, and then Team A and Team B work together for 10 months to complete the task exactly. The construction company needs to pay a construction fee of 80,000 yuan per month to Team A and 50,000 yuan per month to Team B.  
$(1)$ How many months does Team B need to work alone to complete the task?  
$(2)$ To ensure that the project is completed within two years and to minimize costs, the construction company decides to have both Team A and Team B work simultaneously on the project and divide the project into two parts. Team A takes $a$ months to complete one part of the project, and Team B takes $b$ months to complete the other part. It is known that Team A's construction time does not exceed 6 months, Team B's construction time does not exceed 24 months, and $a$ and $b$ are positive integers. How many ways can Team A and Team B actually schedule their construction time? Which scheduling method results in the lowest cost?

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