
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
Tumara needs to travel from city D to city E, then from city E to city F, and then from city F back to city D. The cities form a right-angled triangle as shown, with city F at a distance of 4000 km from D and city E at a distance of 4500 km from D. Travel costs $0.20 per km by bus and, for airplanes, a $150 booking fee plus $0.15 per km. Assume a right angle at F.

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