
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
The pricing rules for "Didi Express Premium" in Changchun City are as follows: the fare consists of mileage fee + time fee:<br/>

|Mileage Fee (by time period)||
|---|---|
|Normal Period (excluding the following 4 periods)|$2.20$ yuan per kilometer|
|$00:00-06:30$|$2.80$ yuan per kilometer|
|$06:30-09:00$|$2.75$ yuan per kilometer|
|$16:00-18:30$|$2.75$ yuan per kilometer|
|$23:00-00:30$|$2.80$ yuan per kilometer|

|Time Fee (by time period)||
|---|---|
|Normal Period (excluding the following 2 periods)|$0.38$ yuan per minute|
|$06:30-09:00$|$0.47$ yuan per minute|
|$16:00-18:30$|$0.47$ yuan per minute|

$(1)$ Xiao Liu took a "Didi Express Premium" to school at $6:10$ in the morning, with a travel distance of $6$ kilometers and a travel time of $10$ minutes. How much should he pay for the fare?<br/>
$(2)$ After school, Xiao Liu took a "Didi Express Premium" home. He got on the car at $16:45$ at school, took another route due to traffic congestion, and the average speed was $30$ kilometers per hour. Let the travel distance on this alternative route be $x$ kilometers $\left(x \leq 30\right)$.<br/>
① How much should Xiao Liu pay for the fare? (Express in algebraic expression with $x$)<br/>
② When $x=8$, calculate how much Xiao Liu should pay for the fare.

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