
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 quadratic function $f(x)=ax^{2}+bx+1$ ($a,b \in \mathbb{R}$), $x \in \mathbb{R}$.  
$(1)$ If the minimum value of the function $f(x)$ is $f(-1)=0$, find the expression for $f(x)$ and state its intervals of monotonicity;  
$(2)$ Under the condition of $(1)$, if $f(x) > x+k$ holds true in the interval $[-3,-1]$, find the range of values for $k$.

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