
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 the function f(x) = ax^2 + 4x + b,

1. Find the range of values for 'a' when b=2, such that f(x) >= 0 holds true for all x in the interval [1, 2].
2. Given that a > b, if f(x) >= 0 holds true for all real numbers x, and there exists an x₀ ∈ R such that ax₀^2 + 4x₀ + b = 0, find the minimum value of $\frac{a^2 + b^2}{a - b}$.

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