
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 $C$: $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1 (a > b > 0)$ has an eccentricity of $\dfrac{\sqrt{2}}{2}$, and a circle centered at the origin with the radius equal to the semi-minor axis of the ellipse is tangent to the line $x - y + \sqrt{2} = 0$.  
$(1)$ Find the equation of the ellipse $C$;  
$(2)$ If a line passing through the point $M(2,0)$ intersects the ellipse $C$ at points $A$ and $B$, and let $P$ be a point on the ellipse such that $\overrightarrow{OA} + \overrightarrow{OB} = t \overrightarrow{OP}$ (where $O$ is the origin), when $|\overrightarrow{PA} - \overrightarrow{PB}| < \dfrac{2\sqrt{5}}{3}$, find the range of the real number $t$.

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