
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
Ket  $f(x) = x^{2} +ax + b$ . If for all nonzero real  $x$   $$ f\left(x + \dfrac{1}{x}\right) = f\left(x\right) + f\left(\dfrac{1}{x}\right) $$  and the roots of  $f(x) = 0$  are integers, what is the value of  $a^{2}+b^{2}$ ?

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