
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 an ellipse with a minor axis length of $2\sqrt{3}$, and the coordinates of the foci are $(-1,0)$ and $(1,0)$.  
$(1)$ Find the standard equation of this ellipse;  

$(2)$ If the line $y=x+m$ intersects this ellipse at two distinct points, find the range of values for $m$.

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