
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 $f(x)=x^{2}+bx+c (b,c \in \mathbb{R}, b < 0)$.
(1) If the domain of $f(x)$ is $[0,1]$ and its range is also $[0,1]$, find the values of $b$ and $c$.
(2) If $b=-2$, find the range of the real number $c$ such that the function $g(x)=\frac{f(x)}{x}$ is always greater than $c$ for any $x \in [3,5]$.

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