
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 largest positive  $p$  ( $p>1$ ) such, that  $\forall a,b,c\in[\frac1p,p]$  the following inequality takes place
\[9(ab+bc+ca)(a^2+b^2+c^2)\geq(a+b+c)^4\]

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