
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
Two locations, A and B, are $800$ km apart. A truck travels from location A to location B at a constant speed not exceeding $100$ km/h. The cost of transporting goods per hour (in yuan) consists of variable costs and fixed costs: the variable cost is $\frac{1}{4}$ times the square of the speed $v$ (in km/h), and the fixed cost is $a$ yuan.  
$(1)$ Express the total transportation cost $y$ (in yuan) as a function of the speed $v$ (in km/h) and state the domain of this function.  
$(2)$ In order to minimize the total transportation cost, at what speed should the truck travel? Calculate the minimum value of the total transportation cost.

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