
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
To reduce energy loss during cooling in summer and heating in winter, the roofs and exterior walls of houses need to be constructed with insulation layers. A building is to be constructed with an insulation layer that can be used for \\(20\\) years, with a construction cost of \\(60,000\\) yuan per centimeter thickness. The annual energy consumption cost \\(C\\) (in ten thousand yuan) of the building is related to the thickness \\(x\\) (in cm) of the insulation layer by the equation \\(C(x) = \frac{k}{3x+5} (0 \leqslant x \leqslant 10)\\), and if no insulation layer is built, the annual energy consumption cost is \\(80,000\\) yuan. Let \\(f(x)\\) be the sum of the construction cost of the insulation layer and the energy consumption cost over \\(20\\) years.

\\((\\)I\\()\\) Find the value of \\(k\\) and the expression for \\(f(x)\\).

\\((\\)II\\()\\) At what thickness of the insulation layer does the total cost \\(f(x)\\) reach its minimum, and what is the minimum value?

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