
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
A certain high school in Harbin city is constructing a thermal insulation layer on the roof and outer walls of the teaching building during the summer vacation to reduce energy loss during winter heating. The thermal insulation layer being built this time can be used for 30 years, and the construction cost of each centimeter thick insulation layer is 80,000 yuan. Due to factors such as construction technology and consumables, the relationship between the annual energy consumption cost $T$ (in units of 10,000 yuan) and the thickness $x$ (in units of cm) of the insulation layer on the teaching building is as follows: When $0\leqslant x\leqslant 5$, $T\left(x\right)=\frac{k}{{3x+4}}$; when $5 \lt x\leqslant 10$, $T\left(x\right)=\frac{1}{{60}}({{x^2}-30x+235})$; If no insulation layer is built, the annual energy consumption cost is 50,000 yuan. Let $f\left(x\right)$ be the sum of the construction cost of the insulation layer and the energy consumption cost for 30 years.

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

$(2)$ When constructing the insulation layer to a certain thickness, at what thickness does the total cost $f\left(x\right)$ reach a minimum? Find the minimum value.

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