
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 natural numbers \( n \) such that for all positive numbers \( a, b, c \) satisfying the inequality

\[ n(ab + bc + ca) > 5(a^2 + b^2 + c^2) \]

there exists a triangle with sides \( a, b, c \).

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