
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 food factory plans to build employee dormitories. The factory uses radiation to sterilize and disinfect food, so the dormitories need radiation protection. The material and quantity of the radiation-proof materials are related to the distance between the dormitory and the factory. The relationship between the cost $p$ (in ten thousand dollars) of building such a dormitory and the distance $x$ (in kilometers) between the dormitory and the factory is: $p=\frac{1000}{x+5}(2≤x≤8)$. For transportation convenience, a simple path needs to be built between the factory and the dormitory. The cost of road construction is 5 ten thousand dollars per kilometer, and the factory provides a one-time transportation subsidy for employees of $\frac{1}{2}({{x}^{2}}+25)$ ten thousand dollars. Let $f(x)$ be the sum of the dormitory construction, road modification fees, and the subsidy provided to employees.

(1) Find the expression for $f(x)$;

(2) Where should the dormitory be built to minimize the total cost $f(x)$, and find the minimum value.

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