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

(I) If $x \in [0,1]$,
(i) Find the range of the function $f(x)$ when $a=b=1$.
(ii) Find the values of $a$ and $b$ if the range of the function $f(x)$ is $[0,1]$.

(II) If $|x|\geqslant 2$, the function $f(x)$ is always non-negative, and the maximum value of $f(x)$ on 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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