
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: $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ ($a > b > 0$) where the minimum distance from a moving point on the ellipse to the foci is $\sqrt{2} - 1$. A circle centered at the origin with the semi-minor axis of the ellipse as its radius is tangent to the line $x - y + \sqrt{2} = 0$.

(Ⅰ) Find the equation of ellipse C;

(Ⅱ) If a line passing through point M(2, 0) intersects ellipse C at points A and B, and point P on the ellipse satisfies $\overrightarrow{OA} + \overrightarrow{OB} = t \overrightarrow{OP}$ (O is the coordinate origin). When $|AB| = \frac{2\sqrt{5}}{3}$, find the value of the real number $t$.

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