
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 maximum value of the positive real number  $k$  such that the inequality  $$ \frac{1}{kab+c^2} +\frac{1} {kbc+a^2} +\frac{1} {kca+b^2}  \geq \frac{k+3}{a^2+b^2+c^2}   $$  holds for all positive real numbers  $a,b,c$  such that  $a^2+b^2+c^2=2(ab+bc+ca).$ 

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