
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, a building's roof and exterior walls need to be insulated. A building is to be fitted with an insulation layer that will last 20 years, with a construction cost of 60,000 RMB per cm in thickness. The building's annual energy consumption cost $C$ (in units of ten thousand RMB) is related to the thickness $x$ (in cm) of the insulation layer by the function $C(x)= \frac{k}{3x+5}$ (for $0 \leq x \leq 10$), and the annual energy consumption cost without insulation is 80,000 RMB. 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) Determine the thickness of the insulation layer that minimizes the total cost $f(x)$ and find the minimum value.

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