
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 $f(x) = ax^2 - 2\sqrt{4 + 2b - b^2}x$, $g(x) = -\sqrt{1 - (x - a)^2}$, where $a, b \in \mathbb{R}$,

(Ⅰ) When $b = 0$, if $f(x)$ is monotonically increasing on $[2, +\infty)$, find the range of values for $a$;

(Ⅱ) Find all pairs of real numbers $(a, b)$ that satisfy the following conditions: when $a$ is an integer, there exists $x_0$, such that $f(x_0)$ is the maximum value of $f(x)$, and $g(x_0)$ is the minimum value of $g(x)$.

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