
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 $f(x)=ax^{2}+bx+c$, it is known that $|f(x)|\leqslant 1$ always holds when $|x| \leqslant 1$.
(I) If $a=1$ and $b=c$, find the range of real number $b$;
(II) If $g(x)=|cx^{2}-bx+a|$, find the maximum value of $g(x)$ when $|x| \leqslant 1$.

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