
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
A courier needs to drive to a certain location within a specified time. If the courier drives at a speed of $1.2$ km per minute, they will arrive $10$ minutes early; if they drive at a speed of $0.8$ km per minute, they will be $5$ minutes late. Find the specified time and the total distance the courier traveled. 

Ranran and Hanhan wrote the following equations:  
Ranran: $1.2(x-10)=0.8(x+5)$  
Hanhan: $\frac{x}{1.2}+10=\frac{x}{0.8}-5$  

$(1)$ In the equation written by Ranran, $x$ represents ______;  
In the equation written by Hanhan, $x$ represents ______;  
$(2)$ Choose one person's method and write out the complete solution process.

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