
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 any positive integer  $n$ , let  $f(n)$  be the maximum number of groups formed by a total of  $n$  people such that the following holds: every group consists of an even number of members, and every two groups share an odd number of members. Compute  $\textstyle\sum_{n=1}^{2022}f(n)\text{ mod }1000$ .

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