
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
a) Find all real numbers  $k$  such that there exists a positive constant  $c_k$  satisfying   $$ (x^2 + 1)(y^2 + 1)(z^2 + 1) \ge c_k(x + y + z)^k $$   for all positive real numbers. 

b) Given the numbers  $k$  found, determine the largest number  $c_k$ .

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