
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 company cafeteria purchased a total of $320$ items of bottled water and vegetables, with bottled water exceeding vegetables by $80$ items.<br/>$(1)$ Find the number of items for bottled water and vegetables respectively;<br/>$(2)$ Now, the plan is to rent a total of $8$ trucks of type A and B to transport all the bottled water and vegetables back to the company cafeteria. It is known that each type A truck can carry up to $40$ items of bottled water and $10$ items of vegetables, while each type B truck can carry up to $20$ items of both bottled water and vegetables. How many possible arrangements are there for the company to use type A and B trucks? Please help design them;<br/>$(3)$ Under the conditions of $(2)$, if each type A truck incurs a transportation cost of $400$ yuan and each type B truck incurs a transportation cost of $360$ yuan. Which arrangement should the company choose to minimize the transportation cost? What is the minimum transportation cost?

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