You are planning a trip through Europe, which includes a visit to six different cities $($Barcelona$,$ London, Paris, Istanbul, Venice, and Prague$).$ The table below includes the one$-$way cost of traveling between the cities. For example, the cost of traveling from Istanbul to Venice is $$500.$ Assume that the cost of traveling from city A to city B is the same as the cost from city B to city A$.$ You must end your trip at the same city where you started your trip. London Paris Istanbul Venice Prague

Barcelona $$1,000$ $$1,200$ $$1,500$ $$1,400$ $$1,700$

London $$300$ $$500$ $$400$ $$450$

Paris $$400$ $$500$ $$325$

Istanbul $$500$ $$275$

Venice $$750$

Given the above travel costs, calculate the total cost of your trip if you visit each city exactly once and return to the starting city.

$(1)$ What is the total cost? And what is the cheapest route to visit all cities?

$2)$ You've heard from fellow travelers that the food on the flight from London to Venice is terrible, and you'd prefer to avoid that experience. With this in mind, what is the revised route and total cost of your trip if you exclude the flight from London to Venice from your itinerary?

$3)$ What is the impact on cost if you must start the journey in Venice?

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