
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 $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1 (a > b > 0)$ with eccentricity $e = \dfrac{\sqrt{2}}{2}$, and it passes through the point $\left( 1, \dfrac{\sqrt{2}}{2} \right)$.

$(1)$ Find the equation of the ellipse;

$(2)$ A line passing through the point $P(0,2)$ intersects the ellipse at two distinct points $M$ and $N$. Find the range of the $x$-axis intercept of the perpendicular bisector of segment $MN$.

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