
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 $D$ to $E$, then from $E$ to $F$, then from $F$ to $D$. 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 $F$ a distance of $4000\text{ km}$ from $D$ and with $E$ a distance of $4500\text{ km}$ from $D$. To take a bus, it costs Piravena $\$0.20$ per kilometer. To take an airplane, it costs her a $\$120$ booking fee, plus $\$0.12$ per kilometer. To begin her trip she flew from $D$ to $E$. Determine the cost to fly from $D$ to $E$.

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