
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 which integers $k$ does there exist a function $f: \mathbb{N} \rightarrow \mathbb{Z}$ satisfying $f(2006) = 2007$ and:

$$
f(xy) = f(x) + f(y) + kf(\operatorname{PGCD}(x, y))
$$

for all integers $x$ and $y$?

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