You’re Stranded on a Desert Island (not dessert)

2. Pot - for boiling water, cooking food
3. Tent - you could make a shelter over time, but having the tent for shelter would allow you to immediately focus on fire, water, and food, and also provide extra bug and rain protection when you sleep at night
4. Rope - you could break it down to make twine to make your own fishing pole, or a fishing net, and also use it to make a bow drill for making fire and snares to catch small animals

What good is a water purifier without a pot to put the purified water in?
You can use heated rocks and hollowed out bamboo or log to boil the water. I've even seen it boiled in a plastic bottle. Most food won't require a pot as you won't have access to grains etc needing it, so a stick and fire will suffice.

A temp shelter until you get more is easy enough to build if you know what you are looking for in nature such as a cliff, fallen trees, and leaves.

Fishing pole would be a waste of your time. Learn to build traps and let the fish swim right into them. That frees you up for building the shelter and processing water\food.