
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 a quadratic function $f\left(x\right)=ax^{2}+bx+1(a$,$b\in R$ and $a\neq 0)$, $x\in R$.<br/>$(1)$ If the minimum value of the function $f\left(x\right)$ is $f\left(1\right)=0$, find the analytical expression of $f\left(x\right)$ and write down the monotonic intervals;<br/>$(2)$ Under the condition of $(1)$, if $f\left(x\right) \gt x+k$ always holds in the interval $\left[1,3\right]$, find the range of values for $k$.

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