
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 ellipse $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ with eccentricity $\frac{\sqrt{2}}{2}$, and the curve passes through the point $(1, \frac{\sqrt{2}}{2})$  
$(1)$ Find the equation of the ellipse $C$;  
$(2)$ Given that the line $x-y+m=0$ intersects the ellipse $C$ at two distinct points $A$ and $B$, and the midpoint of segment $AB$ is not inside the circle $x^2+y^2= \frac{5}{9}$, find the range of values for $m$.

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