
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 all positive integers  $ k$  such that for any positive numbers  $ a, b$  and  $ c$  satisfying the inequality 

\[ k(ab \plus{} bc \plus{} ca) > 5(a^2 \plus{} b^2 \plus{} c^2),\]

there must exist a triangle with  $ a, b$  and  $ c$  as the length of its three sides respectively.

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