
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 quadratic function $f(x)=ax^{2}+bx+c$.
(1) If $f(x)$ satisfies $f(-1-x)=f(-1+x)$ for any $x$, and $f(0)=1$, $f(x)_{min}=0$, find the analytic expression of $f(x)$;
(2) If $a=1$, $c=0$, and $|f(x)|\leqslant 1$ always holds in the interval $(0,1]$, find the range of $b$.

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