
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 $a$, $b$, and $c$ be distinct integers where $b$ is the average of $a$ and $c$, i.e., $b = \frac{a+c}{2}$. Given that
\[\frac{a}{b-c} + \frac{b}{c-a} + \frac{c}{a-b} = 0,\]
find the value of
\[\frac{a}{(b-c)^3} + \frac{b}{(c-a)^3} + \frac{c}{(a-b)^3}.\]

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