
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 needs to travel from city $D$ to city $E$, then from $E$ to city $F$, and finally from $F$ back to city $D$. The cities form a right-angled triangle with city $F$ being 4000 km directly east from city $D$ and city $E$ being 4500 km directly northeast of city $D$. The costs are unchanged: a bus costs $\$0.15$ per km and a flight costs a $\$100$ booking fee plus $\$0.10$ per km. Determine the total distance Piravena travels on her trip.

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