
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 decides to invest $3200$ yuan to build a warehouse (in the shape of a rectangular prism) with a constant height. The back wall will be built reusing the old wall at no cost, the front will be made of iron grilles at a cost of $40$ yuan per meter in length, and the two side walls will be built with bricks at a cost of $45$ yuan per meter in length. The top will have a cost of $20$ yuan per square meter. Let the length of the iron grilles be $x$ meters and the length of one brick wall be $y$ meters. Find:<br/>$(1)$ Write down the relationship between $x$ and $y$;<br/>$(2)$ Determine the maximum allowable value of the warehouse area $S$. In order to maximize $S$ without exceeding the budget, how long should the front iron grille be designed?

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