
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
Let  $f(x,y)$  be a function defined for all pairs of nonnegative integers  $(x, y),$  such that  $f(0,k)=f(k,0)=2^k$  and \[f(a,b)+f(a+1,b+1)=f(a+1,b)+f(a,b+1)\] for all nonnegative integers  $a, b.$  Determine the number of positive integers  $n\leq2016$  for which there exist two nonnegative integers  $a, b$  such that  $f(a,b)=n$ .

*Proposed by Michael Ren*

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