
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 comprehensively promote rural revitalization, a certain province implements a policy of urban assistance to towns. City $A$ has 120 tons of goods, and city $B$ has 130 tons of goods. After investigation, it was found that town A in the province needs 140 tons of goods, and town B needs 110 tons of goods. It was decided that cities $A$ and $B$ would be responsible for assisting towns A and B. It is known that the transportation cost of goods from city $A$ to towns A and B is 300 yuan per ton and 150 yuan per ton, respectively, and the transportation cost of goods from city $B$ to towns A and B is 200 yuan per ton and 100 yuan per ton, respectively.  
$(1)$ Let $x$ tons of goods be transported from city $A$ to town A. The total transportation cost of goods from cities $A$ and $B$ to towns A and B is $y$ yuan. Find the analytical expression of $y$ in terms of $x$.  
$(2)$ Design a transportation plan with the lowest transportation cost and calculate the minimum cost.

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