
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]


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To reduce energy loss during summer cooling and winter heating, the roof and exterior walls of a building need to have insulation installed. A building is to construct an insulation layer that can last for 20 years, where the construction cost per centimeter thickness of the insulation is 60,000 yuan. The annual energy consumption cost C (in ten thousand yuan) and the thickness of the insulation layer x (in cm) satisfy the relationship: $C(x) = \frac{40}{3x + 5}$ (where $1 \leq x \leq 10$). Let $f(x)$ be the sum of the insulation construction cost and energy consumption cost over 20 years.

(Ⅰ) Determine the expression for $f(x)$.

(Ⅱ) Determine the thickness of the insulation layer that minimizes the total cost $f(x)$ and calculate the minimum value.

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