
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
Let  $S$  be the set of all ordered triples  $\left(a,b,c\right)$  of positive integers such that  $\left(b-c\right)^2+\left(c-a\right)^2+\left(a-b\right)^2=2018$  and  $a+b+c\leq M$  for some positive integer  $M$ . Given that  $\displaystyle\sum_{\left(a,b,c\right)\in S}a=k$ , what is \[\displaystyle\sum_{\left(a,b,c\right)\in S}a\left(a^2-bc\right)\] in terms of  $k$ ?

*2018 CCA Math Bonanza Lightning Round #4.1*

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