
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 City $Y$ to City $Z$, and then back from City $Z$ to City $X$. The cities form a right-angled triangle, with $Z$ a distance of 4000 km from $X$ and with $Y$ a distance of 4500 km from $X$. To take a bus, it costs Piravena $\$0.20$ per kilometer. To take an airplane, it costs her a $\$120$ booking fee, plus $\$0.12$ per kilometer.

Determine the distance she travels for her complete trip and decide the cheapest way to complete the journey if Piravena wants to minimize her travel expenses.

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