
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 house 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 per meter of colored steel plate is $450$ yuan, and the unit price per meter of composite steel plate is $200$ yuan. The ground of the house does not require 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 $32000$ yuan.  
$(1)$ Let the length of the front wall be $x$ (meters), and the length of the side walls be $y$ (meters). The material cost required to build a set of houses is $P$ (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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