
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 freight truck travels at a constant speed of $x$ kilometers per hour for $130$ kilometers, where $50 \leq x \leq 100$ (unit: kilometers per hour). Assuming the price of gasoline is $8$ yuan per liter, and the truck consumes $(2+\frac{{x}^{2}}{360})$ liters of gasoline per hour, and the driver's wage is $80$ yuan per hour.  
$(1)$ Find the expression of the total cost $y$ of this trip in terms of $x$.  
$(2)$ At what value of $x$ is the total cost of this trip minimized, and what is the minimum cost?

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