
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
(This question is worth 12 points) A unit decides to invest 3200 yuan to build a warehouse (in the shape of a rectangular prism), with a constant height. Its back wall uses an old wall at no cost, the front is made of iron bars costing 40 yuan per meter, the two side walls are made of bricks costing 45 yuan per meter, and the roof costs 20 yuan per square meter. Calculate:
(1) What is the maximum allowable value of the warehouse area $S$?
(2) To maximize $S$ without exceeding the budget, how long should the front iron bars be designed?

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