
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  $f$  be a function defined on the set of positive integers , and with values in the same set, which satisfies: $\bullet$   $f (n + f (n)) = 1$  for all  $n\ge 1$ . $\bullet$   $f (1998) = 2$ Find the lowest possible value of the sum  $f (1) + f (2) +... + f (1999)$ , and find the formula of  $f$  for which this minimum is satisfied,

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