
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
Piravena is making a triangular trip starting from city $X$ to city $Y$, from $Y$ to city $Z$, and returning to $X$. The cities form a right-angled triangle with $Z$ being the right angle. The distance from $X$ to $Z$ is 4000 km, and from $X$ to $Y$ is 5000 km. The cost of taking the bus is $\$0.20$ per kilometer and to take an airplane is a $\$120$ booking fee plus $\$0.12$ per kilometer. Determine the total distance Piravena travels for her round trip.

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