
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  $m, n$  are positive integers and  $gcd(m, n) = 1$ , find  $100m + n$ .


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