
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 $M:\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1 (a > b > 0)$ has an eccentricity of $\dfrac{\sqrt{2}}{2}$, and one of its foci is at $(\sqrt{2},0)$.

$(I)$ Find the standard equation of the ellipse $M$;

$(II)$ If the line $y=x+m$ intersects the ellipse $M$ at points $A$ and $B$, find the maximum area of $\Delta OAB$ ($O$ is the origin).

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