
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
This year, Harbin is undergoing a gas engineering renovation project, which is jointly contracted by two engineering teams, Team A and Team B. If Team A works alone, it will take 10 days to complete the project; if Team B works alone, it will take 15 days to complete.<br/>$(1)$ If Teams A and B work together for 4 days, and the remaining work is completed by Team B, how many more days does Team B need to complete the task?<br/>$(2)$ Under the conditions of $(1)$, if the remuneration given to the two engineering teams is distributed in proportion to the amount of work completed, and the total remuneration for this engineering renovation project is 100,000 yuan, how much money will Team A and Team B each receive?

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