
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
Didi Express is a convenient means of transportation, and the pricing rules are as follows:

| Pricing Item | Mileage Fee | Time Fee | Long-distance Fee |
|--------------|-------------|----------|-------------------|
| Unit Price   | $1.8$ yuan per kilometer | $0.45$ yuan per minute | $0.4$ yuan per kilometer |

Note: The fare consists of three parts: mileage fee, time fee, and long-distance fee. The mileage fee is calculated based on the actual mileage traveled; the time fee is calculated based on the actual time traveled; the long-distance fee is charged as follows: within 10 kilometers (including 10 kilometers), no long-distance fee is charged. For distances exceeding 10 kilometers, an additional $0.4$ yuan per kilometer is charged.

$(1)$ If Xiao Dong takes a Didi Express, travels 5 kilometers, and the travel time is 10 minutes, how much does he need to pay for the fare?

$(2)$ If Xiao Ming takes a Didi Express, travels $a$ kilometers, and the travel time is $b$ minutes, how much should Xiao Ming pay for the fare? (Express in terms of $a$ and $b$, and simplify)

$(3)$ Xiao Wang and Xiao Zhang each take a Didi Express, with travel distances of 9.5 kilometers and 14.5 kilometers, respectively. Xiao Wang's travel time is 24 minutes longer than Xiao Zhang's. Calculate and explain the relationship between the fares paid by the two when they get off.

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