
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
Construct a pigpen with a ground area of $12m^2$ that backs onto a wall. The front of the rectangular pigpen costs $120$ yuan per square meter to build, the sides cost $80$ yuan per square meter, and the roof costs $1120$ yuan. If the wall height is $3m$, and the cost of the back wall is not considered, what design minimizes the total cost of the pigpen, and what is the minimum total cost in yuan?

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