
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 inequality \( m(a^3 + b^3 + c^3) \geq 6(a^2 + b^2 + c^2) + 1 \) holds.

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