
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 $X$ to $Y$, then from $Y$ to $Z$, and finally from $Z$ to $X$. The cities form a right-angled triangle, with $Z$ a distance of 4000 km from $X$ and $Y$ a distance of 4500 km from $X$. Traveling by bus costs $\$0.20$ per kilometer, and traveling by airplane costs a $\$120$ booking fee plus $\$0.12$ per kilometer. Determine the total distance Piravena travels for her complete trip.

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