
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]


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The highway management bureau studied the relationship between the flow velocity $v$ (unit: kilometers per hour) and the flow density $x$ (unit: vehicles per kilometer) on the highway during the National Day holiday: $v=\left\{\begin{array}{l}{100, 0<x\leq 10}\\{150-\frac{k}{160-x}, 10<x\leq 110}\end{array}\right.$, where $k \in \mathbb{R}$. The study showed that when the flow density reaches 110 vehicles per kilometer, it causes severe congestion, and at this point, the flow velocity is 0 kilometers per hour.  
$(1)$ If the flow velocity $v$ is not less than 90 kilometers per hour, find the range of values for the flow density $x$;  
$(2)$ The flow rate $y$ (number of vehicles passing in a unit of time, unit: vehicles per hour) satisfies $y=x\cdot v$. Find the maximum flow rate on the highway during the National Day holiday (rounded to 1 vehicle per hour), and indicate the flow density when the flow rate is at its maximum (rounded to 1 vehicle per kilometer). (Given: $\sqrt{5} \approx 2.236$)

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