
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
During the wheat harvest season, to ensure that the wheat grains are stored in the granary, a farm arranges for two types of harvesters, $A$ and $B$, to harvest wheat. It is known that one $A$ harvester harvests $2$ hectares more wheat on average per day than one $B$ harvester. The time it takes for one $A$ harvester to harvest $15$ hectares of wheat is the same as the time it takes for one $B$ harvester to harvest $9$ hectares of wheat.<br/>$(1)$ How many hectares of wheat does one $A$ harvester and one $B$ harvester each harvest on average per day?<br/>$(2)$ It is known that the fee for using an $A$ harvester is $50$ yuan per hectare, and the fee for using a $B$ harvester is $45$ yuan per hectare. The farm arranges for a total of $12$ harvesters of both types to harvest wheat simultaneously. In order to ensure that at least $50$ hectares of wheat are harvested each day, how many $A$ harvesters should be arranged to minimize the cost? What is the minimum cost?

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