
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 truck travels at a constant speed of $x$ kilometers per hour for a distance of 300 kilometers. According to traffic regulations, $50 \leq x \leq 100$ (unit: kilometers per hour). Assuming the price of gasoline is $8$ yuan per liter, the truck consumes $(4+\frac{x^2}{420})$ liters of gasoline per hour, and the driver's wage is $50$ yuan per hour.  
$(1)$ Find the expression of the total cost $y$ (in yuan) of this trip in terms of $x$ (kilometers per hour).  
$(2)$ For what value of $x$ does the total cost $y$ of this trip reach its minimum? Calculate the value of the minimum cost. (All results should be accurate to $1$ yuan).

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