
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 transportation company has two types of trucks, $A$ and $B$. One $A$ truck and two $B$ trucks can transport $50$ tons of goods at a time, while five $A$ trucks and four $B$ trucks can transport $160$ tons of goods at a time.  
$(1)$ Question: How many tons of goods can each type of truck transport at a time?  
$(2)$ Currently, there are $190$ tons of goods that need to be transported. The company plans to arrange the $A$ and $B$ trucks to transport all the goods at once (with each truck fully loaded). The transportation cost for an $A$ truck is $500$ yuan, and for a $B$ truck is $400$ yuan. Please list all the transportation plans and indicate which plan has the lowest cost.

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