
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
For all positive integers  $n$ ,  $k$ , let  $f(n, 2k)$  be the number of ways an  $n \times 2k$  board can be fully covered by  $nk$  dominoes of size  $2 \times 1$ . (For example,  $f(2, 2)=2$  and  $f(3, 2)=3$ .) Find all positive integers  $n$  such that for every positive integer  $k$ , the number  $f(n, 2k)$  is odd.

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