
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
The "Green Leaf" housekeeping service company dispatched 16 cleaners to clean the rooms of the newly renovated "Sea Sky" hotel. There are two sizes of rooms, large and small. Each cleaner can clean 4 large rooms or 5 small rooms in a day. If $x$ people are assigned to clean the large rooms, the rest will clean the small rooms. The payment for cleaning a large room is 80 yuan, and for a small room is 60 yuan.  
$(1)$ Write the function relationship between the company's daily income $y$ (in yuan) and the number of people $x$.  
$(2)$ How should the 16 cleaners be arranged to generate a daily income of 5000 yuan for the "Green Leaf" housekeeping service company?

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