
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 plans to build a batch of simple houses (the house type is rectangular, with a height of 2.5 meters). The front and back walls use 2.5-meter high color steel plates, and the two sides use 2.5-meter 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 uniformly 2.5 meters, and the price of a steel plate is the length of the steel plate multiplied by the unit price). The unit price per meter is 450 yuan for color steel plates and 200 yuan for composite steel plates. The roof uses other materials, with a material cost of 200 yuan per square meter. The material cost for each house is controlled within 32,000 yuan.
1. Let the length of the front wall of the house be x, the length of the two sides be y, and the material cost be p. Express p in terms of x and y.
2. Under the control of material costs, what is the maximum value of the area S of the simple house? And indicate the length x of the front wall when S is the largest.

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