
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
To transport 240 tons of fresh vegetables from location A to location B, there are three types of vehicles available: A, B, and C. The carrying capacity and freight cost of each vehicle are shown in the table below: (assuming each vehicle is fully loaded)

| Vehicle Type | A   | B   | C   |
|--------------|-----|-----|-----|
| Carrying Capacity (tons/vehicle) | 10  | 16  | 20  |
| Freight Cost (元/vehicle)       | 800 | 1000| 1200|

$(1)$ To transport all the vegetables, 8 vehicles of type A and 5 vehicles of type B are needed. The number of vehicles of type C needed is ______.

$(2)$ If all the vegetables are transported using vehicles of type A and B, and the total freight cost is 16400元, how many vehicles of type A and B are needed respectively?

$(3)$ In order to save on freight costs, vehicles of type A, B, and C are planned to be used simultaneously for transportation. It is known that the total number of vehicles is 16. Can you find the number of vehicles for each type? What is the total freight cost in this case?

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