
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 $a,b \in \mathbb{R}$:

1. Find the expression for the minimum value of the function $f(x)$ on $[-1,1]$, denoted as $g(a)$, when $b=\frac{a^{2}}{4}+1$.
2. If the function $f(x)$ has a zero point on $[-1,1]$ and $0 \leqslant b-2a \leqslant 1$, find the range of $b$.

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