
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
Given that the distance between locations $A$ and $B$ is $100$ km. According to traffic regulations, the speed of vehicles on the road between the two locations should be limited to $\left[50,100\right]$ km/h, and the fuel price is $8$ yuan/L. Assuming that the fuel consumption rate of a car traveling at a speed of $x$ km/h is $(4+\frac{{x}^{2}}{400})$ L/h, and the driver's labor cost is $40$ yuan/h.  
$(Ⅰ)$ Express the total cost $W$ as a function of the speed $x$.  
$(Ⅱ)$ Determine the value of $x$ that minimizes the total cost $W$.

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