
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 construction site decided to build a batch of simple houses with a rectangular shape and a height of 2.5 meters. The front and back walls of the houses are made of 2.5 meters high colored steel plates, and the two side walls are made of 2.5 meters high composite steel plates. The prices of both types of steel plates are calculated by length (i.e., the height of the steel plates is 2.5 meters. The price of a steel plate is calculated by multiplying its length by the unit price). It is known that the unit price of colored steel plates is 450 yuan per meter, and the unit price of composite steel plates is 200 yuan per meter. The ground of the house does not need additional materials, and the roof is made of other materials, costing 200 yuan per square meter. The material cost for each set of houses is controlled within 32,000 yuan.
(1) Let the length of the front wall of the house be $x$ (meters), the length of the side walls be $y$ (meters), and the material cost for building a set of houses be $P$ (yuan). Try to express $P$ in terms of $x$ and $y$;
(2) Try to find the maximum value of the area $S$ of a simple house. What should the length of the front wall be when $S$ is at its maximum?

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