
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
When studying the traffic situation in a certain city, road density refers to the number of vehicles passing through a certain section of road divided by time, and vehicle density is the number of vehicles passing through a certain section of road divided by the length of that section. The traffic flow is defined as $v=\frac{q}{x}$, where $x$ is the road density and $q$ is the vehicle density. The traffic flow is given by $v=f\left(x\right)=\left\{\begin{array}{l}{100-135\cdot(\frac{1}{3})^{\frac{80}{x}},0<x<40}\\{-k(x-40)+85,40\leq x\leq 80}\end{array}\right.$.

$(1)$ If the traffic flow $v \gt 95$, find the range of values for the road density $x$.

$(2)$ Given that the road density $x=80$ and the measured traffic flow $v=50$, find the maximum value of the vehicle density $q$.

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