
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
Find the smallest real number \( m \) such that for any positive real numbers \( a, b, c \) satisfying \( a + b + c = 1 \), the following inequality holds:
\[ m\left(a^{3} + b^{3} + c^{3}\right) \geq 6\left(a^{2} + b^{2} + c^{2}\right) + 1. \]

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