
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 must make a trip from $A$ to $B$, then from $B$ to $C$, then from $C$ to $A$. Each of these three parts of the trip is made entirely by bus or entirely by airplane. The cities form a right-angled triangle as shown, with $C$ a distance of 4000 km from $A$ and with $B$ a distance of 4500 km from $A$. To take a bus, it costs Piravena $\$0.20$ per kilometer. To take an airplane, it costs her a $\$120$ booking fee, plus $\$0.15$ per kilometer. For the segment $BC$, the airplane costs an additional $\$50$ due to extra fees. Determine the least expensive way to travel between each city and the total cost.

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