
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) Determine the largest real number  $A$  with the following property: For all non-negative real numbers  $x,y,z$ , one has
\[\frac{1+yz}{1+x^2}+\frac{1+zx}{1+y^2}+\frac{1+xy}{1+z^2} \ge A.\]
b) For this real number  $A$ , find all triples  $(x,y,z)$  of non-negative real numbers for which equality holds in the above inequality.

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