
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
Before the Spring Festival in the Year of the Ox, the Vegetable King Xiao Ming plans to transport a batch of vegetables to be sold in other places. If he uses $2$ type $A$ trucks and $1$ type $B$ truck to carry a full load of vegetables at a time, he can transport $10$ tons. If he uses $1$ type $A$ truck and $2$ type $B$ trucks to carry a full load of vegetables at a time, he can transport $11$ tons. There are $31$ tons of vegetables available. Xiao Ming plans to rent $x$ type $A$ trucks and $y$ type $B$ trucks at the same time to transport all the vegetables in one go, with each truck carrying a full load of vegetables. Based on the information above, answer the following questions:  
$(1)$ How many tons can $1$ type $A$ truck and $1$ type $B$ truck carry when fully loaded?  
$(2)$ Please help design a rental plan for the logistics company.  
$(3)$ If the rental fee for $1$ type $A$ truck is $100$ yuan per trip, and the rental fee for $1$ type $B$ truck is $120$ yuan per trip, select the rental plan with the lowest cost and calculate the minimum rental fee.

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