
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
In order to reduce energy loss during cooling in summer and heating in winter, the roof and outer walls of a building need to be constructed with insulation layers. A certain building needs to build an insulation layer that can be used for 15 years. The construction cost of each centimeter thick insulation layer is 80,000 yuan. The annual energy consumption cost $C$ (unit: 10,000 yuan) of the building is related to the thickness $x$ (unit: cm) of the insulation layer by the function $C(x) = \frac{k}{3x+5}$, where $0 \leq x \leq 10$. 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 for 15 years.

$(1)$ Find the value of $k$ and the expression of $f(x)$.

$(2)$ 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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