
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  $f:\mathbb Z\rightarrow \mathbb Z$  such that, for all integers  $a,b,c$  that satisfy  $a+b+c=0$ , the following equality holds:
\[f(a)^2+f(b)^2+f(c)^2=2f(a)f(b)+2f(b)f(c)+2f(c)f(a).\]
(Here  $\mathbb{Z}$  denotes the set of integers.)

*Proposed by Liam Baker, South Africa*

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