
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 needs to travel from city $X$ to city $Y$, then from $Y$ to city $Z$, and finally from $Z$ back to $X$. Each segment of her journey is completed entirely by bus or by airplane. The cities form a right-angled triangle with city $Z$ as the right angle. The distance from $X$ to $Z$ is 4000 km and from $X$ to $Y$ is 5000 km. Bus fare is $\$0.20$ per kilometer, and airplane travel costs a $\$120$ booking fee plus $\$0.12$ per kilometer. Piravena chooses the least expensive travel option for each segment. Determine the total cost of her trip.

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