
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 concept of "low-carbon living, green travel" has gradually been accepted by most people. A certain bicycle specialty store has two types of bicycles, $A$ and $B$, with profits of $a$ yuan per bike for type $A$ and $b$ yuan per bike for type $B$. The sales situation of this specialty store in the first two weeks of October is as follows:

|           | Sales of Type $A$ (bikes) | Sales of Type $B$ (bikes) | Total Profit (yuan) |
|-----------|---------------------------|---------------------------|---------------------|
| Week 1    | 10                        | 12                        | 2000                |
| Week 2    | 20                        | 15                        | 3100                |

$(1)$ Find the values of $a$ and $b$;  
$(2)$ If in the third week, a total of 25 bicycles of types $A$ and $B$ were sold, with the sales of type $B$ exceeding the sales of type $A$ but not exceeding twice the sales of type $A$, how many bikes of type $A$ and type $B$ should the store sell to maximize profit in the third week, and what is the maximum profit in yuan?

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