
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
Lorena needs to travel from city $D$ to city $E$, then from city $E$ to city $F$, and then back to city $D$. The cities form a right-angled triangle, with $D$ a distance of 4000 km from $F$ and $E$ a distance of 4500 km from $F$. Traveling by bus costs Lorena $\$0.20$ per kilometer. Traveling by airplane costs a $\$120$ booking fee, plus $\$0.12$ per kilometer. Lorena chooses the least expensive travel option for each segment.

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