
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, Shanghai recently required the construction of insulation layers on the roofs and exterior walls of new residential buildings. A building needs to construct an insulation layer that can be used for 20 years, with a construction cost of 60,000 yuan per centimeter of 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 as $C(x) = (0 \leq x \leq 10)$. 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.
(1) Find the value of $k$ and the expression for $f(x)$;
(2) 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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