
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
Solve the system of linear equations application problem.<br/>On December 18, 2023, a magnitude 6.2 earthquake occurred in Gansu. The emergency and transportation departments of Liaoning province provided strong assistance. A comprehensive inspection was conducted on the safety of houses, power supply, and material support in the disaster area. Now, two types of trucks, A and B, are arranged to transport supplies from a pharmaceutical company warehouse to the earthquake-stricken area. The situation of the two types of trucks is shown in the table below:

|            | Truck A (units) | Truck B (units) | Total (tons) |
|------------|-----------------|-----------------|--------------|
| First time | 3               | 4               | 27           |
| Second time| 4               | 5               | 35           |

$(1)$ How many tons of goods can each truck of type A and type B carry?<br/>$(2)$ It is known that in this transportation, each truck is fully loaded. Truck A costs $100$ yuan per ton of goods transported, and truck B costs $150 yuan per ton of goods transported. There are 5 trucks participating in the transportation, with $a$ trucks of type A. Find the functional relationship between the total cost $w$ of the trucks and $a$.<br/>$(3)$ Under the conditions of $(2)$, how should the goods be arranged to minimize the total cost? What is the minimum total cost?

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