
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
The positive integer-valued function  $f(n)$  satisfies  $f(f(n)) = 4n$  and  $f(n + 1) > f(n) > 0$  for all positive integers  $n$ . Compute the number of possible 16-tuples  $(f(1), f(2), f(3), \dots, f(16))$ .

*Proposed by Lewis Chen*

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