
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 function of $x$, $y=(m+6)x^{2}+2(m-1)x+m+1$, always has a root.  
$(1)$ Find the range of $m$;  
$(2)$ If the function has two distinct roots, and the sum of their reciprocals is $-4$, find the value of $m$.

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