
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's annual electricity bill is $240,000. In order to save energy and reduce emissions, the company decided to install a solar power supply device that can be used for 15 years and connected to the company's power grid. The installation of such a power supply device requires a one-time investment of a fixed cost $G$ (in units of $10,000) which is proportional to the installation area $x$ of the solar panels, with a proportionality factor of $k=0.5$. The company estimates that the annual electricity bill $C$ (in units of $10,000) after installation is related to the installation area $x$ of the solar panels by the function $C(x)=\frac{b}{{20x+100}}$ (where $x\geqslant 0$, and $b$ is a constant). If the solar panels are not installed, the annual electricity bill will still be $240,000. Let $F$ be the sum of the fixed cost $G$ and the 15-year electricity bill.

$(1)$ Find the value of the constant $b$, and calculate the annual electricity bill $C$ for the company after installing 10 square meters of solar panels.

$(2)$ Establish the functional relationship of $F$ with respect to $x$.

$(3)$ After installing how many square meters of solar panels, will $F$ reach its minimum value? What is the minimum value in units of $10,000?

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