
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
Teverin needs to travel from city D to city E, then from city E to city F, and finally from city F back to city D. The cities form a right-angled triangle with D as the right angle. The distance from D to E is 4000 km and from D to F is 4200 km. Traveling by bus costs $\$0.20$ per kilometer. Traveling by airplane involves a $\$120$ booking fee plus $\$0.12$ per kilometer. Teverin chooses the least expensive mode of travel for each segment.

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