
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 $f(x)=x^{3}+ax^{2}+bx+a^{2}$ where $a,b\in\mathbb{R}$.
(1) If the function $f(x)$ has an extreme value of $10$ at $x=1$, find the value of $b$.
(2) If for any $a\in[-4,+\infty)$, $f(x)$ is monotonically increasing on $x\in[0,2]$, find the minimum value of $b$.

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