
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
A certain car rental company wants to purchase a total of $10$ vehicles, including sedans and vans. They must purchase at least $3$ sedans. Each sedan costs $70,000, and each van costs $40,000. The company can invest up to $550,000 in purchasing vehicles. 

$(1)$ How many purchasing plans meet the company's requirements? Please explain the reasoning.

$(2)$ If the daily rental fee for each sedan is $200 and for each van is $110, and assuming that all 10 newly purchased vehicles can be rented out daily, in order to ensure that the daily rental income from these 10 vehicles is not less than $1,500, which purchasing plan should be chosen?

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