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private void InvokeAndComplete (Action<T> act, TaskCompletionSource<byte> promise)
{
 try {
 act (internalValue);
 } finally {
 // some value we will ignore
 promise.SetResult (1);
 }
}
public Task Alter (Action<T> alterAction)
{
 TaskCompletionSource<byte> promise = new TaskCompletionSource<byte> ();
 Task localLastTask;
 do {
 // Not sure if Volatile.Read is necessary
 localLastTask = Volatile.Read (ref lastTask);
 } while (Interlocked.CompareExchange (ref lastTask, promise.Task, localLastTask) != localLastTask);
 return localLastTask.ContinueWith ((currentTask) => {
 InvokeAndComplete (alterAction, promise);
 }, taskScheduler);
}

private void InvokeAndComplete (Action<T> act, TaskCompletionSource<byte> promise)
{
 try {
 act (internalValue);
 } finally {
 // some value we will ignore
 promise.SetResult (1);
 }
}
public Task Alter (Action<T> alterAction)
{
 TaskCompletionSource<byte> promise = new TaskCompletionSource<byte> ();
 Task localLastTask;
 do {
 // Not sure if Volatile.Read is necessary
 localLastTask = Volatile.Read (ref lastTask);
 } while (Interlocked.CompareExchange (ref lastTask, promise.Task, localLastTask) != localLastTask);
 // only this thread has a reference to the object referenced by localLastTask
 return localLastTask.ContinueWith ((currentTask) => {
 InvokeAndComplete (alterAction, promise);
 }, taskScheduler);
}

Doing this now

private void InvokeAndComplete (Action<T> act, TaskCompletionSource<byte> promise)
{
 try {
 act (internalValue);
 } finally {
 // some value we will ignore
 promise.SetResult (1);
 }
}
public Task Alter (Action<T> alterAction)
{
 TaskCompletionSource<byte> promise = new TaskCompletionSource<byte> ();
 Task localLastTask;
 do {
 // Not sure if Volatile.Read is necessary
 localLastTask = Volatile.Read (ref lastTask);
 } while (Interlocked.CompareExchange (ref lastTask, promise.Task, localLastTask) != localLastTask);
 return localLastTask.ContinueWith ((currentTask) => {
 InvokeAndComplete (alterAction, promise);
 }, taskScheduler);
}

• I'd like to be able to ContinueWith the Task task directly, instead of having to start it within a different submitted Action. I feel like I need to create the new Task so that I can atomically exchange the lastTask field's value.
• Lambda capturing of local variables, partly alleviated with state objects where possible.
• How can I keep a reference to the currently executing Task, so I can short-circuit execution? Or should I use a poison pill to signal termination and not chain any new tasks?
• Interlocked for atomicity. Should I use a ConcurrentQueue? How would I manage not having a dedicated thread and only running one Task at a time with a ConcurrentQueue?

• I'd like to be able to ContinueWith the Task task directly, instead of having to start it within a different submitted Action. I feel like I need to create the new Task so that I can atomically exchange the lastTask field's value.
• Lambda capturing of local variables, partly alleviated with state objects where possible.
• How can I keep a reference to the currently executing Task, so I can short-circuit execution? Or should I use a poison pill to signal termination and not chain any new tasks?
• Interlocked for atomicity. Should I use a ConcurrentQueue? How would I manage not having a dedicated thread and only running one Task at a time with a ConcurrentQueue?
• Is a Volatile.Read of lastTask necessary for localCurrent = lastTask;?