
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 \( a > b \), the quadratic inequality \( ax^{2}+2x+b \geqslant 0 \) holds for all real numbers \( x \), and there exists \( x_{0} \in \mathbb{R} \) such that \( ax_{0}^{2}+2x_{0}+b=0 \) is satisfied. Find the minimum value of \( 2a^{2}+b^{2} \).

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