
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
The one-variable quadratic equation about $x$ is $x^{2}+2x+3-k=0$ with two distinct real roots.<br/>$(1)$ Find the range of $k$;<br/>$(2)$ If the two roots of the equation are $\alpha $ and $\beta $, and $k^{2}=\alpha \beta +3k$, find the value of $k$.

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