
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, it is necessary to construct insulation layers on the roofs and exterior walls of buildings. A building needs to construct an insulation layer that can be used for 20 years, with a construction cost of 30,000 yuan per centimeter of thickness. The annual energy consumption cost $C$ (unit: ten thousand yuan) of the building is related to the thickness $x$ (unit: cm) of the insulation layer by the equation $C(x)= \frac {k}{3x+5}(0\leqslant x\leqslant 10)$. If no insulation layer is built, the annual energy consumption cost is 4 ten thousand 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 $f(x)$.
$(2)$ When the thickness of the insulation layer is how much, the total cost $f(x)$ is minimized, and find the minimum value.

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