
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  $f_0$  be the function from  $\mathbb{Z}^2$  to  $\{0,1\}$  such that  $f_0(0,0)=1$  and  $f_0(x,y)=0$  otherwise. For each positive integer  $m$ , let  $f_m(x,y)$  be the remainder when \[ f_{m-1}(x,y) + \sum_{j=-1}^{1} \sum_{k=-1}^{1} f_{m-1}(x+j,y+k) \] is divided by  $2$ .
Finally, for each nonnegative integer  $n$ , let  $a_n$  denote the number of pairs  $(x,y)$  such that  $f_n(x,y) = 1$ .
Find a closed form for  $a_n$ .

*Proposed by Bobby Shen*

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