
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 plans another trip involving cities $D$, $E$, and $F$, forming a right-angled triangle. City $F$ is 3500 km from $D$ directly, and $D$ is 4000 km from $E$. The cost of taking a bus is now $\$0.20$ per kilometer. The cost of taking an airplane is a $\$120$ booking fee for flights up to 3000 km and a $\$150$ booking fee for flights over 3000 km, plus $\$0.12$ per kilometer. Piravena wants to choose the least expensive travel options for her trip segments: from $D$ to $E$, from $E$ to $F$, and from $F$ to $D$.

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