
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 that the foci of ellipse $C$ are $F\_1(-1,0)$ and $F\_2(1,0)$, and point $P\left(-1, \frac{ \sqrt{2}}{2}\right) $ is on ellipse $C$.

(I) Find the equation of ellipse $C$;

(II) If parabola ${{y}^{2}}=2px(p > 0)$ intersects with ellipse $C$ at points $M$ and $N$, when the area of $\Delta OMN$ (where $O$ is the coordinate origin) takes the maximum value, find the value of $p$.

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