
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 an ellipse $(C)$: $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)$, which passes through the point $(\sqrt{2}, 1)$, and a triangle formed by the two endpoints of the ellipse's minor axis and one focus is an isosceles right triangle.
(I) Find the standard equation of the ellipse;
(II) Let $M(x, y)$ be a moving point on the ellipse $(C)$, and $P(p, 0)$ be a fixed point on the x-axis. Find the minimum value of $|MP|$ and the coordinates of the point $M$ when the minimum value is reached.

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