
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
Let   $x$  and  $y$  are  real numbers such that   $x^2y^2+2yx^2+1=0.$   If   $S=\frac{2}{x^2}+1+\frac{1}{x}+y(y+2+\frac{1}{x})$ ,  find  
(a)max $S$    and  
(b) min $S$ .

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