
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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