Updating a local variable within new threads

Question

I am working on designing a multi-user application and would like to have a singleton service class that is shared amongst all of the user threads.

Within this class, I would like to have a method that does some heavy lifting to eventually compile a seperate list for each user, I am under the impression that since this is a method, each user thread will recieve its own sperate copy of this on its call stack.

Since this process is time consuming, I would like create multiple threads within the method to handle the processing and add items to a single list that was declared and instantiated within the method.

In the end I am envisioning a list complete list for each user that will then be able to be used throughout the rest of the application within the users thread.

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My question is: How do accomplish sharing the list declared in the method amongst the multiple threads I want to create within the method while maintaining thread safety?

The flow chart below may explain everything better