
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 $g(x)=x-1$.

$(1)$ If there exists $x \in \mathbb{R}$ such that $f(x) < b \cdot g(x)$, find the range of the real number $b$;  
$(2)$ Let $F(x)=f(x)-mg(x)+1-m$. If $F(x) \geqslant 0$ holds true in the interval $[2,5]$, find the range of the real number $m$.

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