
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 functions  $f:\mathbb{N} \to \mathbb{N}$  so that for any distinct positive integers  $x,y,z$  the value of  $x+y+z$  is a perfect square if and only if  $f(x)+f(y)+f(z)$  is a perfect square.

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