
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
Find all integers  $k$  for which,  there is a function  $f: N \to Z$  that satisfies:
(i)  $f(1995) = 1996$ (ii)  $f(xy) = f(x) + f(y) + kf(m_{xy})$  for all natural numbers  $x, y$ ,where $ m_{xy}$  denotes the greatest common divisor of the numbers  $x, y$ .

Clarification:  $N = \{1,2,3,...\}$  and  $Z = \{...-2,-1,0,1,2,...\}$  .

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