
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
Driving under the influence of alcohol is a serious violation that endangers traffic safety. According to relevant national regulations, a driver with an alcohol content of $20\sim 79mg$ in $100mL$ of blood is considered to be driving under the influence, while a driver with $80mg$ or more is considered to be driving while intoxicated. Assuming that after drinking a certain amount of alcohol, if a person stops drinking, the alcohol content in the blood will decrease at a rate of $p$ per hour. Drivers A, B, and C have all consumed a certain amount of alcohol, and their blood alcohol content has all risen to $1mg/mL$. (Operational process is rounded to $4$ decimal places. Reference data: $\lg 2\approx 0.3010$, $\lg 3\approx 0.4771$, $\lg 7\approx 0.8451$, $\frac{1}{{\sqrt[6]{5}}}≈0.7647$, $\frac{1}{{\sqrt[7]{5}}}≈0.7946$)

$(1)$ If driver A stops drinking and the rate of decrease in blood alcohol content per hour is $p_{1}=30\%$, how many hours at least does driver A need to legally drive? (Round the final result to the nearest integer)

$(2)$ Driver B drives after stopping drinking for $5$ hours but is still considered to be driving under the influence. Based on the calculation in $(1)$, provide a simple analysis from a mathematical perspective for the reason behind this and suggest a reasonable solution for B to be able to drive legally.

$(3)$ After listening to your analysis, driver B, without changing the amount of alcohol consumed, tests his blood alcohol content after stopping drinking for $6$ and $7$ hours respectively and records the results. After observing for a period of time, B finds that he needs at least $7$ hours to be able to drive legally. Help B estimate the range of values for the rate of decrease in blood alcohol content per hour after stopping drinking. (Round the final result to two decimal places)

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