
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 the number of ordered triples of integers  $(a,b,c)$  such that   $$ a^2 + b^2 + c^2  - ab - bc - ca - 1 \le 4042b - 2021a - 2021c - 2021^2 $$   and  $|a|, |b|, |c| \le 2021.$ 

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