
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 factory uses radiation to sterilize food and is now planning to build a dormitory for its workers near the factory, with radiation protection measures for the dormitory. The choice of radiation protection materials for the building and the distance of the dormitory from the factory are related. If the total cost of building the dormitory $p$ (in ten thousand yuan) and the distance $x$ (in km) from the dormitory to the factory is given by: $p= \dfrac{1000}{x+5} (2\leqslant x\leqslant 8)$. For convenience of transportation, a simple access road will also be built between the factory and the dormitory, with the cost of building the road being 5 ten thousand yuan per kilometer, and the factory provides a one-time subsidy for the workers' transportation costs of $\dfrac{1}{2}(x^{2}+25)$ ten thousand yuan. Let $f(x)$ be the sum of the costs of building the dormitory, the road construction, and the subsidy given to the workers.  
$(1)$ Find the expression for $f(x)$;  
$(2)$ How far should the dormitory be built from the factory to minimize the total cost $f(x)$, and what is the minimum value?

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