
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
Car rental companies A and B each have 50 cars for rent. The following is a conversation between the two managers:<br/>A company manager: If the monthly rent for each car in my company is $3000, then all 50 cars can be rented out. If the monthly rent for each car increases by $50, one less car will be rented out. In addition, the company pays a monthly maintenance fee of $200 for each rented car.<br/>B company manager: The monthly rent for each car in my company is $3500. Regardless of whether cars are rented out or not, the company must pay a total monthly maintenance fee of $1850.

Explanation: 
1. The number of cars is an integer.
2. Monthly profit = Monthly rent fee - Monthly maintenance fee.
3. The difference in monthly profit between the two companies = Profit of the company with higher profit - Profit of the company with lower profit.

Under the condition that the number of cars rented out by both companies is equal, solve the following problems:
(1) When each company rents out 10 cars, the monthly profit of company A is ______ yuan; when each company rents out ____ cars, the monthly profit of both companies is equal.
(2) Find the maximum value of the difference in monthly profits between the two companies.
(3) Company A is enthusiastic about public welfare and donates $a$ yuan ($a > 0$) to a charity for each car rented out. If after the donation, the remaining monthly profit of company A is still higher than the monthly profit of company B, and when both companies rent out 17 cars each, the difference between the remaining monthly profit of company A and the monthly profit of company B is the greatest. Find the range of values for $a$.

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