
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 $f(x) = x^2 + ax + b$ ($a, b \in \mathbb{R}$).

(Ⅰ) Given $x \in [0, 1]$,

(i) If $a = b = 1$, find the range of the function $f(x)$;

(ii) If the range of the function $f(x)$ is $[0, 1]$, find the values of $a$ and $b$;

(Ⅱ) When $|x| \geq 2$, it always holds that $f(x) \geq 0$, and the maximum value of $f(x)$ in the interval $(2, 3]$ is 1, find the maximum and minimum values of $a^2 + b^2$.

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