
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
The expression $27x^6 - 512y^6$ can be factored as $(ax+by)(cx^2+dxy+ey^2)(fx+gy)(hx^2+jxy+ky^2)$. If $a$, $b$, $c$, $d$, $e$, $f$, $g$, $h$, $j$, and $k$ are all integers, find their sum.

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