
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 solid) with a fixed height. The rear wall utilizes the old wall at no cost, the front is made of iron fences costing 40 yuan per meter, the two side walls are made of brick at a cost of 45 yuan per meter, and the top costs 20 yuan per square meter.
Calculate:
(1) What is the maximum allowable value for the area $S$ of the warehouse?
(2) To maximize $S$ without exceeding the budget, how long should the iron fence in the front be designed?

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