
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 ordered triples of non-negative integers  $(a,b,c)$  such that  $a^2+2b+c$ ,  $b^2+2c+a$ , and  $c^2+2a+b$  are all perfect squares.

*Proposed by Matthew Babbitt*

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