
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 the functions  $f: \mathbb{Z}\to \mathbb{Z}$  satisfying the following property: if  $a$ ,  $b$  and  $c$  are integers such that  $a+b+c=0$ , then  $$ f(a)+f(b)+f(c)=a^2+b^2+c^2. $$  

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