
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 $f(x)=ax^{2}+bx+1$, where $a,b \in \mathbb{R}$, the function $f(x)$ attains its minimum value at $x=-1$, and the minimum value is $0$.
(1) Find the analytic expression of $f(x)$;
(2) The equation $f(x)=|x+1|-k+3$ has exactly two distinct real roots with respect to $x$. Find the range of the real number $k$.

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