
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
Given \( f(u) = u^{2} + au + (b-2) \), where \( u = x + \frac{1}{x} \) (with \( x \in \mathbb{R} \) and \( x \neq 0 \)). If \( a \) and \( b \) are real numbers such that the equation \( f(u) = 0 \) has at least one real root, find the minimum value of \( a^{2} + b^{2} \).

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