
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
In 2023, the specific charging standards for fuel taxis in Beijing continue to use the charging standards of 2022 as follows:  
① Taxis charge a starting price of $13$ yuan for trips within $3$ kilometers, plus an additional $1$ yuan fuel surcharge. For distances exceeding $3$ kilometers, the excess is charged at a rate of $2.30$ yuan per kilometer (less than $1$ kilometer is charged as $1$ kilometer).  
② Reservation service fee: For reservations made more than $4$ hours in advance, the fee is $6$ yuan per reservation; for reservations made within $4 hours, the fee is $5$ yuan per reservation.  
③ For one-way trips exceeding $15$ kilometers, an additional $50\%$ of the original fare rate ($2.30$ yuan) is charged.  
④ Taxi fare settlement is in yuan, rounded to the nearest yuan (rounding up for amounts less than $1$ yuan).  
(If the fare is less than the starting price, the starting price should be charged.)  
Read the following information:  
On March 24, 2023, the first-year students of Sanfan Middle School are going to the West City Maternal and Child Health Hospital for a physical examination. In the morning, after getting up at $6:40$, Xiaogao, a student, prepared to reserve a taxi to go to the examination center at $12:00$ noon from Sanfan Middle School. The navigation estimated the fare to be $43$ yuan. One hour before departure, he found that the alley was congested, which was not conducive to the taxi's passage. Therefore, he decided to walk to the intersection $600$ meters ahead on the way, rebook a taxi to the examination center, and found that the estimated fare was $42$ yuan when he rebooked (the two routes were the same and unobstructed). Because the end time of the examination could not be determined, on the return journey, Xiaogao did not book a taxi in advance, but chose to call a taxi. On the way back from the examination center to Sanfan Middle School, when he was $2$ kilometers away from the school, he found that his clothes were left at the examination center. He immediately asked the taxi driver to turn back and return to the examination center to get off.  
Based on the above information, answer the following questions:  
$(1)$ Given that the distance from Sanfan Middle School to the examination center is $x$ kilometers, what is the range of $x$?  
$(2)$ After Xiaogao returns to the examination center, how much money should he pay to the driver?

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