
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 certain unit is planning to build a rectangular room that is backed against a wall with a ground surface area of 12 square meters. Due to geographical constraints, the side length x of the room cannot exceed 5 meters. The cost of constructing the front face of the room is 400 yuan per square meter, the cost for the sides is 150 yuan per square meter, and the total cost for the roof and ground is 5800 yuan. If the wall height is 3 meters, and the cost of the back face is disregarded, at what side length will the total construction cost be minimized? What is the minimum total construction cost?

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