
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
In times of trouble, help comes from all directions. A certain city government has raised 120 tons of essential drought relief supplies to be transported to the disaster area. There are three types of vehicles, A, B, and C, available for selection. The carrying capacity and transportation costs of each vehicle are shown in the table below: (assuming each vehicle is fully loaded)

| Type of Vehicle | A   | B   | C   |
|-----------------|-----|-----|-----|
| Carrying Capacity (tons/vehicle) | $5$ | $8$ | $10$ |
| Transportation Cost (元/vehicle) | $300$ | $400$ | $500$ |

$(1)$ If all the supplies are to be transported using types A and B vehicles, and the transportation cost is 6400 yuan, how many vehicles of types A and B are needed respectively?

$(2)$ In order to save on transportation costs, the city government decides that at least two of the three vehicle types, A, B, and C, will be involved in transportation. It is known that the total number of vehicles is 18. By setting up a system of equations, find the quantities of the three types of vehicles.

$(3)$ Determine which transportation plan saves the most on transportation costs and how much is saved.

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