
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 store plans to purchase 36 desks, each valued at 20 yuan, in batches within a month. Each batch purchases $x$ desks ($x$ is a positive integer), and each batch requires a shipping fee of 4 yuan. The storage fee for storing the purchased desks for a month is proportional to the total value of the desks purchased per batch (excluding shipping fees). If 4 desks are purchased per batch, the total shipping and storage fees for the month will be 52 yuan. Now, there are only 48 yuan available for the month to cover shipping and storage fees.  
(1) Find the total cost $f(x)$ of shipping and storage fees for the month;  
(2) Is it possible to properly arrange the quantity of each batch of goods so that the funds are sufficient? Write your conclusion and explain your reasoning.  
(3) To minimize the funds used for shipping and storage fees for the month, how many desks should be purchased per batch?

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