
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 $C$: $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1 (a > b > 0)$ passes through point $P(\sqrt{6},1)$, and its left focus is $F(-2,0)$.  
$(1)$ Find the equation of ellipse $C$;  
$(2)$ If the line $y=x+m$ intersects ellipse $C$ at two distinct points $A$ and $B$, and the midpoint of line segment $AB$, denoted as $M$, lies on the circle $x^2+y^2=1$, find the value of $m$.

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