
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 recent years, a certain company consumes about $240,000$ yuan of electricity per year. In order to save energy and reduce emissions, it decides to install a solar power supply device that can be used for $15$ years and connect it to the company's power grid. The installation cost (in units of 10,000 yuan) of this type of power supply device is directly proportional to the area of the solar panels (in square meters), with a proportion coefficient of about $0.5$. To ensure normal power consumption, after installation, a mode of complementary power supply using both solar energy and electricity is adopted. Assuming that under this mode, the function relationship between the annual electricity consumption $C$ (in units of 10,000 yuan) after installation and the area $x$ (in square meters) of the installed solar panels is $C(x)=\dfrac{k}{20x+100}$ (for $x\geqslant 0$, where $k$ is a constant). Let $F$ be the sum of the cost of installing this type of solar power supply device and the electricity consumption over $15$ years.

(1) Explain the practical meaning of $C(0)$ and establish the functional relationship of $F$ with respect to $x$.

(2) For what value of $x$ in square meters does $F$ achieve the minimum value? What is the minimum value in yuan?

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