
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 functions $f(x)=x^{2}$ and $g(x)=x-1$.
(1) Find the range of the real number $b$ such that there exists an $x \in \mathbb{R}$ that satisfies $f(x) < b \cdot g(x)$.
(2) Let $F(x)=f(x)-mg(x)+1-m-m^{2}$, and $|F(x)|$ is monotonically increasing on $[0,1]$. Find the range of the real number $m$.

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