
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 unit is constructing a rectangular, ground-level room with an area of $12 m^{2}$ that is backed against a wall. Due to geographical constraints, the length $x$ (in meters) of the side of the house cannot exceed $a$ meters. The construction cost for the front of the house is $400$ yuan$/m^{2}$, and the cost for the sides is $150$ yuan$/m^{2}$. The combined cost for the roof and floor is $5800$ yuan. The height of the wall is $3$ meters, with no cost considered for the back of the house.

(1) Represent the total construction cost $y$ (in yuan) as a function of $x$, and write out its domain.
(2) What should be the length of the side when the total construction cost of the house is the lowest? What is the minimum total cost?

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