
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 \( f(x) = x^3 + 3x - 1 \) have roots \( a, b, c \). Given that
\[
\frac{1}{a^3 + b^3} + \frac{1}{b^3 + c^3} + \frac{1}{c^3 + a^3}
\]
can be written as \( \frac{m}{n} \), where \( m \) and \( n \) are positive integers and \( \operatorname{gcd}(m, n) = 1 \), find \( 100m + n \).

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